Theorem cbvmo 2117

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 1802	. . 3 |- ( E. x ph <-> E. y ps )
5	1, 2, 3	cbveu 2101	. . 3 |- ( E! x ph <-> E! y ps )
6	4, 5	imbi12i 239	. 2 |- ( ( E. x ph -> E! x ph ) <-> ( E. y ps -> E! y ps ) )
7		df-mo 2081	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
8		df-mo 2081	. 2 |- ( E* y ps <-> ( E. y ps -> E! y ps ) )
9	6, 7, 8	3bitr4i 212	1 |- ( E* x ph <-> E* y ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1506   E.wex 1538   E!weu 2077   E*wmo 2078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081

This theorem is referenced by:  cbvmow  2118  dffun6f  5331