Theorem cbvrex 1795

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-17 969	. . . 4 |- (x e. A -> A.y x e. A)
2		cbvral.1	. . . 4 |- (ph -> A.yph)
3	1, 2	hban 1007	. . 3 |- ((x e. A /\ ph) -> A.y(x e. A /\ ph))
4		ax-17 969	. . . 4 |- (y e. A -> A.x y e. A)
5		cbvral.2	. . . 4 |- (ps -> A.xps)
6	4, 5	hban 1007	. . 3 |- ((y e. A /\ ps) -> A.x(y e. A /\ ps))
7		eleq1 1531	. . . 4 |- (x = y -> (x e. A <-> y e. A))
8		cbvral.3	. . . 4 |- (x = y -> (ph <-> ps))
9	7, 8	anbi12d 627	. . 3 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ ps)))
10	3, 6, 9	cbvex 1164	. 2 |- (E.x(x e. A /\ ph) <-> E.y(y e. A /\ ps))
11		df-rex 1647	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
12		df-rex 1647	. 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 952   = wceq 954   e. wcel 956  E.wex 978  E.wrex 1643

This theorem is referenced by:  cbvrexv 1797  cbvrexsv 1964  cbviun 2584  isarep1 3569  elrnopabg 3791  abrexexlem2 3850  elrnoprabg 4114  cau3i 6859

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-cleq 1467  df-clel 1470  df-rex 1647

Copyright terms: Public domain