Theorem cbvmo 2119

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 1804	. . 3 |- ( E. x ph <-> E. y ps )
5	1, 2, 3	cbveu 2103	. . 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 2083	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
8		df-mo 2083	. 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 1508   E.wex 1540   E!weu 2079   E*wmo 2080

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083

This theorem is referenced by:  cbvmow  2120  dffun6f  5339