Theorem cbvmo 1804

Description: Rule used to change bound variables, using implicit substitition. (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 8-Jun-2011.)

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)

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 1529	. . 3 |- (E.xph <-> E.yps)
5	1, 2, 3	cbveu 1785	. . 3 |- (E!xph <-> E!yps)
6	4, 5	imbi12i 205	. 2 |- ((E.xph -> E!xph) <-> (E.yps -> E!yps))
7		df-mo 1776	. 2 |- (E*xph <-> (E.xph -> E!xph))
8		df-mo 1776	. 2 |- (E*yps <-> (E.yps -> E!yps))
9	6, 7, 8	3bitr4i 200	1 |- (E*xph <-> E*yps)

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298  E.wex 1326  E!weu 1771  E*wmo 1772

This theorem is referenced by:  dffun6f 4435

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

Copyright terms: Public domain