Theorem reuxfr2 1579

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A.

Hypotheses

Ref	Expression
reuxfr2.1	|- (y e. B -> A e. B)
reuxfr2.2	|- (x e. B -> E*y(y e. B /\ x = A))

Assertion

Ref	Expression
reuxfr2	|- (E!x e. B E.y e. B (x = A /\ ph) <-> E!y e. B ph)

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

Proof of Theorem reuxfr2

Step	Hyp	Ref	Expression
1		2reuswap 1341	. . . 4 |- (A.x e. B E*y(y e. B /\ (x = A /\ ph)) -> (E!x e. B E.y e. B (x = A /\ ph) -> E!y e. B E.x e. B (x = A /\ ph)))
2		reuxfr2.2	. . . . . 6 |- (x e. B -> E*y(y e. B /\ x = A))
3		moan 1046	. . . . . 6 |- (E*y(y e. B /\ x = A) -> E*y(ph /\ (y e. B /\ x = A)))
4	2, 3	syl 12	. . . . 5 |- (x e. B -> E*y(ph /\ (y e. B /\ x = A)))
5		ancom 333	. . . . . . 7 |- ((ph /\ (y e. B /\ x = A)) <-> ((y e. B /\ x = A) /\ ph))
6		anass 336	. . . . . . 7 |- (((y e. B /\ x = A) /\ ph) <-> (y e. B /\ (x = A /\ ph)))
7	5, 6	bitr 151	. . . . . 6 |- ((ph /\ (y e. B /\ x = A)) <-> (y e. B /\ (x = A /\ ph)))
8	7	bimo 1031	. . . . 5 |- (E*y(ph /\ (y e. B /\ x = A)) <-> E*y(y e. B /\ (x = A /\ ph)))
9	4, 8	sylib 173	. . . 4 |- (x e. B -> E*y(y e. B /\ (x = A /\ ph)))
10	1, 9	mprg 1249	. . 3 |- (E!x e. B E.y e. B (x = A /\ ph) -> E!y e. B E.x e. B (x = A /\ ph))
11		2reuswap 1341	. . . 4 |- (A.y e. B E*x(x e. B /\ (x = A /\ ph)) -> (E!y e. B E.x e. B (x = A /\ ph) -> E!x e. B E.y e. B (x = A /\ ph)))
12		moeq 1431	. . . . . . 7 |- E*x x = A
13	12	moani 1047	. . . . . 6 |- E*x((x e. B /\ ph) /\ x = A)
14		ancom 333	. . . . . . . 8 |- (((x e. B /\ ph) /\ x = A) <-> (x = A /\ (x e. B /\ ph)))
15		an12 370	. . . . . . . 8 |- ((x = A /\ (x e. B /\ ph)) <-> (x e. B /\ (x = A /\ ph)))
16	14, 15	bitr 151	. . . . . . 7 |- (((x e. B /\ ph) /\ x = A) <-> (x e. B /\ (x = A /\ ph)))
17	16	bimo 1031	. . . . . 6 |- (E*x((x e. B /\ ph) /\ x = A) <-> E*x(x e. B /\ (x = A /\ ph)))
18	13, 17	mpbi 164	. . . . 5 |- E*x(x e. B /\ (x = A /\ ph))
19	18	a1i 7	. . . 4 |- (y e. B -> E*x(x e. B /\ (x = A /\ ph)))
20	11, 19	mprg 1249	. . 3 |- (E!y e. B E.x e. B (x = A /\ ph) -> E!x e. B E.y e. B (x = A /\ ph))
21	10, 20	impbi 139	. 2 |- (E!x e. B E.y e. B (x = A /\ ph) <-> E!y e. B E.x e. B (x = A /\ ph))
22		reuxfr2.1	. . . 4 |- (y e. B -> A e. B)
23		pm4.2i 149	. . . . 5 |- (x = A -> (ph <-> ph))
24	23	ceqsrexv 1413	. . . 4 |- (A e. B -> (E.x e. B (x = A /\ ph) <-> ph))
25	22, 24	syl 12	. . 3 |- (y e. B -> (E.x e. B (x = A /\ ph) <-> ph))
26	25	bireua 1319	. 2 |- (E!y e. B E.x e. B (x = A /\ ph) <-> E!y e. B ph)
27	21, 26	bitr 151	1 |- (E!x e. B E.y e. B (x = A /\ ph) <-> E!y e. B ph)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E*wmo 1008   = wceq 1091   e. wcel 1092  E.wrex 1202  E!wreu 1203

This theorem is referenced by:  reuxfr 1580

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-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-v 1349

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