Theorem cbvrex 1802

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:   x,y,A

Proof of Theorem cbvrex

Step	Hyp	Ref	Expression
1		ax-17 973	. . . 4 |- (x e. A -> A.y x e. A)
2		cbvral.1	. . . 4 |- (ph -> A.yph)
3	1, 2	hban 1011	. . 3 |- ((x e. A /\ ph) -> A.y(x e. A /\ ph))
4		ax-17 973	. . . 4 |- (y e. A -> A.x y e. A)
5		cbvral.2	. . . 4 |- (ps -> A.xps)
6	4, 5	hban 1011	. . 3 |- ((y e. A /\ ps) -> A.x(y e. A /\ ps))
7		eleq1 1537	. . . 4 |- (x = y -> (x e. A <-> y e. A))
8		cbvral.3	. . . 4 |- (x = y -> (ph <-> ps))
9	7, 8	anbi12d 630	. . 3 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ ps)))
10	3, 6, 9	cbvex 1168	. 2 |- (E.x(x e. A /\ ph) <-> E.y(y e. A /\ ps))
11		df-rex 1653	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
12		df-rex 1653	. 2 |- (E.y e. A ps <-> E.y(y e. A /\ ps))
13	10, 11, 12	3bitr4 183	1 |- (E.x e. A ph <-> E.y e. A ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  E.wrex 1649

This theorem is referenced by:  cbvrexv 1804  cbvrexsv 1971  cbviun 2593  isarep1 3583  elrnopabg 3806  abrexexlem2 3865  elrnoprabg 4130  cau3i 6914

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-cleq 1472  df-clel 1475  df-rex 1653

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