Theorem cbvmo 1940

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 1639	. . 3 |- ( E. x ph <-> E. y ps )
5	1, 2, 3	cbveu 1924	. . 3 |- ( E! x ph <-> E! y ps )
6	4, 5	imbi12i 228	. 2 |- ( ( E. x ph -> E! x ph ) <-> ( E. y ps -> E! y ps ) )
7		df-mo 1904	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
8		df-mo 1904	. 2 |- ( E* y ps <-> ( E. y ps -> E! y ps ) )
9	6, 7, 8	3bitr4i 201	1 |- ( E* x ph <-> E* y ps )

Syntax hints:   -> wi 4   <-> wb 98   F/wnf 1349   E.wex 1381   E!weu 1900   E*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by:  dffun6f  4915