Theorem zfrepclf 1477

Description: An inference rule based on the Axiom of Replacement. Typically, ph defines a function from x to y.

Hypotheses

Ref	Expression
zfrepclf.1	|- (w e. A -> A.x w e. A)
zfrepclf.2	|- A e. V
zfrepclf.3	|- (x e. A -> E.zA.y(ph -> y = z))

Assertion

Ref	Expression
zfrepclf	|- E.zA.y(y e. z <-> E.x(x e. A /\ ph))

Distinct variable group(s):   y,z,A   ph,z   w,A   x,y,z   x,w

Proof of Theorem zfrepclf

Step	Hyp	Ref	Expression
1		zfrepclf.2	. 2 |- A e. V
2		ax-17 925	. . . . . 6 |- (w e. v -> A.x w e. v)
3		zfrepclf.1	. . . . . 6 |- (w e. A -> A.x w e. A)
4	2, 3	hbeq 1171	. . . . 5 |- (v = A -> A.x v = A)
5		eleq2 1150	. . . . . 6 |- (v = A -> (x e. v <-> x e. A))
6		zfrepclf.3	. . . . . 6 |- (x e. A -> E.zA.y(ph -> y = z))
7	5, 6	syl6bi 187	. . . . 5 |- (v = A -> (x e. v -> E.zA.y(ph -> y = z)))
8	4, 7	19.21ai 740	. . . 4 |- (v = A -> A.x(x e. v -> E.zA.y(ph -> y = z)))
9		ax-17 925	. . . . 5 |- (ph -> A.zph)
10	9	zfrep3 1476	. . . 4 |- (A.x(x e. v -> E.zA.y(ph -> y = z)) -> E.zA.y(y e. z <-> E.x(x e. v /\ ph)))
11	8, 10	syl 12	. . 3 |- (v = A -> E.zA.y(y e. z <-> E.x(x e. v /\ ph)))
12	5	anbi1d 469	. . . . . . 7 |- (v = A -> ((x e. v /\ ph) <-> (x e. A /\ ph)))
13	4, 12	biexd 783	. . . . . 6 |- (v = A -> (E.x(x e. v /\ ph) <-> E.x(x e. A /\ ph)))
14	13	bibi2d 470	. . . . 5 |- (v = A -> ((y e. z <-> E.x(x e. v /\ ph)) <-> (y e. z <-> E.x(x e. A /\ ph))))
15	14	bialdv 935	. . . 4 |- (v = A -> (A.y(y e. z <-> E.x(x e. v /\ ph)) <-> A.y(y e. z <-> E.x(x e. A /\ ph))))
16	15	biexdv 936	. . 3 |- (v = A -> (E.zA.y(y e. z <-> E.x(x e. v /\ ph)) <-> E.zA.y(y e. z <-> E.x(x e. A /\ ph))))
17	11, 16	mpbid 170	. 2 |- (v = A -> E.zA.y(y e. z <-> E.x(x e. A /\ ph)))
18	1, 17	vtocle 1391	1 |- E.zA.y(y e. z <-> E.x(x e. A /\ ph))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  zfrep3cl 1478  zfrep4 1479

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-12 802  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

metamath.org