Theorem cbvreuv 1335

Description: Change the bound variable of a restricted uniqueness quantifier using implicit substitution.

Hypothesis

Ref	Expression
cbvralv.1	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvreuv	|- (E!x e. A ph <-> E!y e. A ps)

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

Proof of Theorem cbvreuv

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 |- ((x e. A /\ ph) -> A.y(x e. A /\ ph))
2		ax-17 925	. . 3 |- ((y e. A /\ ps) -> A.x(y e. A /\ ps))
3		eleq1 1149	. . . 4 |- (x = y -> (x e. A <-> y e. A))
4		cbvralv.1	. . . 4 |- (x = y -> (ph <-> ps))
5	3, 4	anbi12d 476	. . 3 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ ps)))
6	1, 2, 5	cbveu 1018	. 2 |- (E!x(x e. A /\ ph) <-> E!y(y e. A /\ ps))
7		df-reu 1207	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
8		df-reu 1207	. 2 |- (E!y e. A ps <-> E!y(y e. A /\ ps))
9	6, 7, 8	3bitr4 158	1 |- (E!x e. A ph <-> E!y e. A ps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = weq 797  E!weu 1007   e. wcel 1092  E!wreu 1203

This theorem is referenced by:  aceq1 3552  aceq2 3554  uzwo3 4616

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-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-cleq 1097  df-clel 1099  df-reu 1207

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