Theorem cbvmo 1981

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
cbvmo.1	|- F/ y ph
cbvmo.2	|- F/ x ps
cbvmo.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvmo	|- ( E* x ph <-> E* y ps )

Proof of Theorem cbvmo

Step	Hyp	Ref	Expression
1		cbvmo.1	. . . 4 |- F/ y ph
2		cbvmo.2	. . . 4 |- F/ x ps
3		cbvmo.3	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvex 1679	. . 3 |- ( E. x ph <-> E. y ps )
5	1, 2, 3	cbveu 1965	. . 3 |- ( E! x ph <-> E! y ps )
6	4, 5	imbi12i 237	. 2 |- ( ( E. x ph -> E! x ph ) <-> ( E. y ps -> E! y ps ) )
7		df-mo 1945	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
8		df-mo 1945	. 2 |- ( E* y ps <-> ( E. y ps -> E! y ps ) )
9	6, 7, 8	3bitr4i 210	1 |- ( E* x ph <-> E* y ps )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1389   E.wex 1421   E!weu 1941   E*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  dffun6f  4935