Theorem cbveu 1368

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

Hypotheses

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

Assertion

Ref	Expression
cbveu	|- (E!xph <-> E!yps)

Distinct variable group:   x,y

Proof of Theorem cbveu

Step	Hyp	Ref	Expression
1		cbveu.1	. . 3 |- (ph -> A.yph)
2	1	sb8eu 1367	. 2 |- (E!xph <-> E!y[y / x]ph)
3		cbveu.2	. . . 4 |- (ps -> A.xps)
4		cbveu.3	. . . 4 |- (x = y -> (ph <-> ps))
5	3, 4	sbie 1179	. . 3 |- ([y / x]ph <-> ps)
6	5	eubii 1364	. 2 |- (E!y[y / x]ph <-> E!yps)
7	2, 6	bitr 173	1 |- (E!xph <-> E!yps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 950   = wceq 1099  E!weu 1357

This theorem is referenced by:  cbvmo 1385  cbvreuv 1777  euuni 2844  fnopabg 3555  tz6.12f 3677  climeu 6988

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-11o 1202

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359

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