Theorem axrepndlem1 3738

Description: Lemma for the Axiom of Replacement with no distinct variable conditions.

Assertion

Ref	Expression
axrepndlem1	|- (-. A.y y = z -> E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))))

Distinct variable group(s):   x,z   x,y

Proof of Theorem axrepndlem1

Step	Hyp	Ref	Expression
1		axrep 1473	. 2 |- E.x(E.yA.w([w / z]ph -> w = y) -> A.w(w e. x <-> E.x(x e. y /\ A.y[w / z]ph)))
2		eq6 826	. . 3 |- (-. A.y y = z -> A.x -. A.y y = z)
3		eq6 826	. . . . 5 |- (-. A.y y = z -> A.y -. A.y y = z)
4		eq6 826	. . . . . 6 |- (-. A.y y = z -> A.z -. A.y y = z)
5		ax-17 925	. . . . . . . . 9 |- (ph -> A.wph)
6	5	hbsb3 875	. . . . . . . 8 |- ([w / z]ph -> A.z[w / z]ph)
7	6	a1i 7	. . . . . . 7 |- (-. A.y y = z -> ([w / z]ph -> A.z[w / z]ph))
8		nd5 3736	. . . . . . 7 |- (-. A.y y = z -> (w = y -> A.z w = y))
9	4, 7, 8	hbimd 787	. . . . . 6 |- (-. A.y y = z -> (([w / z]ph -> w = y) -> A.z([w / z]ph -> w = y)))
10		sbequ12r 866	. . . . . . . 8 |- (w = z -> ([w / z]ph <-> ph))
11		a8b 817	. . . . . . . 8 |- (w = z -> (w = y <-> z = y))
12	10, 11	imbi12d 474	. . . . . . 7 |- (w = z -> (([w / z]ph -> w = y) <-> (ph -> z = y)))
13	12	a1i 7	. . . . . 6 |- (-. A.y y = z -> (w = z -> (([w / z]ph -> w = y) <-> (ph -> z = y))))
14	4, 9, 13	cbvald 977	. . . . 5 |- (-. A.y y = z -> (A.w([w / z]ph -> w = y) <-> A.z(ph -> z = y)))
15	3, 14	biexd 783	. . . 4 |- (-. A.y y = z -> (E.yA.w([w / z]ph -> w = y) <-> E.yA.z(ph -> z = y)))
16		ax-17 925	. . . . . . 7 |- (w e. x -> A.z w e. x)
17	16	a1i 7	. . . . . 6 |- (-. A.y y = z -> (w e. x -> A.z w e. x))
18		ddeel2 1004	. . . . . . . . 9 |- (-. A.z z = y -> (x e. y -> A.z x e. y))
19	18	eq4ds 823	. . . . . . . 8 |- (-. A.y y = z -> (x e. y -> A.z x e. y))
20	6	hbal 700	. . . . . . . . 9 |- (A.y[w / z]ph -> A.zA.y[w / z]ph)
21	20	a1i 7	. . . . . . . 8 |- (-. A.y y = z -> (A.y[w / z]ph -> A.zA.y[w / z]ph))
22	19, 21	hband 788	. . . . . . 7 |- (-. A.y y = z -> ((x e. y /\ A.y[w / z]ph) -> A.z(x e. y /\ A.y[w / z]ph)))
23	2, 22	hbexd 791	. . . . . 6 |- (-. A.y y = z -> (E.x(x e. y /\ A.y[w / z]ph) -> A.zE.x(x e. y /\ A.y[w / z]ph)))
24	4, 17, 23	hbbid 789	. . . . 5 |- (-. A.y y = z -> ((w e. x <-> E.x(x e. y /\ A.y[w / z]ph)) -> A.z(w e. x <-> E.x(x e. y /\ A.y[w / z]ph))))
25		a13b 819	. . . . . . . 8 |- (w = z -> (w e. x <-> z e. x))
26	25	adantl 305	. . . . . . 7 |- ((-. A.y y = z /\ w = z) -> (w e. x <-> z e. x))
27		ddeeq2 1002	. . . . . . . . . . 11 |- (-. A.y y = z -> (w = z -> A.y w = z))
28	27	imp 277	. . . . . . . . . 10 |- ((-. A.y y = z /\ w = z) -> A.y w = z)
29		hba1 698	. . . . . . . . . . 11 |- (A.y w = z -> A.yA.y w = z)
30	10	a4s 682	. . . . . . . . . . 11 |- (A.y w = z -> ([w / z]ph <-> ph))
31	29, 30	biald 782	. . . . . . . . . 10 |- (A.y w = z -> (A.y[w / z]ph <-> A.yph))
32	28, 31	syl 12	. . . . . . . . 9 |- ((-. A.y y = z /\ w = z) -> (A.y[w / z]ph <-> A.yph))
33	32	anbi2d 468	. . . . . . . 8 |- ((-. A.y y = z /\ w = z) -> ((x e. y /\ A.y[w / z]ph) <-> (x e. y /\ A.yph)))
34	33	biexdv 936	. . . . . . 7 |- ((-. A.y y = z /\ w = z) -> (E.x(x e. y /\ A.y[w / z]ph) <-> E.x(x e. y /\ A.yph)))
35	26, 34	bibi12d 477	. . . . . 6 |- ((-. A.y y = z /\ w = z) -> ((w e. x <-> E.x(x e. y /\ A.y[w / z]ph)) <-> (z e. x <-> E.x(x e. y /\ A.yph))))
36	35	exp 291	. . . . 5 |- (-. A.y y = z -> (w = z -> ((w e. x <-> E.x(x e. y /\ A.y[w / z]ph)) <-> (z e. x <-> E.x(x e. y /\ A.yph)))))
37	4, 24, 36	cbvald 977	. . . 4 |- (-. A.y y = z -> (A.w(w e. x <-> E.x(x e. y /\ A.y[w / z]ph)) <-> A.z(z e. x <-> E.x(x e. y /\ A.yph))))
38	15, 37	imbi12d 474	. . 3 |- (-. A.y y = z -> ((E.yA.w([w / z]ph -> w = y) -> A.w(w e. x <-> E.x(x e. y /\ A.y[w / z]ph))) <-> (E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph)))))
39	2, 38	biexd 783	. 2 |- (-. A.y y = z -> (E.x(E.yA.w([w / z]ph -> w = y) -> A.w(w e. x <-> E.x(x e. y /\ A.y[w / z]ph))) <-> E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph)))))
40	1, 39	mpbii 168	1 |- (-. A.y y = z -> E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803  [wsb 852

This theorem is referenced by:  axrepndlem2 3739

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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