Theorem cbvmo 1401

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

Hypotheses

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

Assertion

Ref	Expression
cbvmo	|- (E*xph <-> E*yps)

Distinct variable group:   x,y

Proof of Theorem cbvmo

Step	Hyp	Ref	Expression
1		cbvmo.1	. . . 4 |- (ph -> A.yph)
2		cbvmo.2	. . . 4 |- (ps -> A.xps)
3		cbvmo.3	. . . 4 |- (x = y -> (ph <-> ps))
4	1, 2, 3	cbvex 1162	. . 3 |- (E.xph <-> E.yps)
5	1, 2, 3	cbveu 1384	. . 3 |- (E!xph <-> E!yps)
6	4, 5	imbi12i 188	. 2 |- ((E.xph -> E!xph) <-> (E.yps -> E!yps))
7		df-mo 1376	. 2 |- (E*xph <-> (E.xph -> E!xph))
8		df-mo 1376	. 2 |- (E*yps <-> (E.yps -> E!yps))
9	6, 7, 8	3bitr4 183	1 |- (E*xph <-> E*yps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   = wceq 953  E.wex 977  E!weu 1373  E*wmo 1374

This theorem is referenced by:  dffunmof 3516

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376

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