Theorem cbvrex 1332

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

Ref	Expression
cbvral.1	|- (ph -> A.yph)
cbvral.2	|- (ps -> A.xps)
cbvral.3	|- (x = y -> (ph <-> ps))

Assertion

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

Distinct variable group(s):   x,y,A

Proof of Theorem cbvrex

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 |- (x e. A -> A.y x e. A)
2		cbvral.1	. . . 4 |- (ph -> A.yph)
3	1, 2	hban 704	. . 3 |- ((x e. A /\ ph) -> A.y(x e. A /\ ph))
4		ax-17 925	. . . 4 |- (y e. A -> A.x y e. A)
5		cbvral.2	. . . 4 |- (ps -> A.xps)
6	4, 5	hban 704	. . 3 |- ((y e. A /\ ps) -> A.x(y e. A /\ ps))
7		eleq1 1149	. . . 4 |- (x = y -> (x e. A <-> y e. A))
8		cbvral.3	. . . 4 |- (x = y -> (ph <-> ps))
9	7, 8	anbi12d 476	. . 3 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ ps)))
10	3, 6, 9	cbvex 849	. 2 |- (E.x(x e. A /\ ph) <-> E.y(y e. A /\ ps))
11		df-rex 1206	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
12		df-rex 1206	. 2 |- (E.y e. A ps <-> E.y(y e. A /\ ps))
13	10, 11, 12	3bitr4 158	1 |- (E.x e. A ph <-> E.y e. A ps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  cbvrexv 1334  elrnopab 2884  abrexexlem2 2911  elrnoprab 3054

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-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-rex 1206

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