Theorem cbvmow 2118

Description: Rule used to change bound variables, using implicit substitution. Version of cbvmo 2117 with a disjoint variable condition. (Contributed by NM, 9-Mar-1995.) (Revised by GG, 23-May-2024.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem cbvmow

Step	Hyp	Ref	Expression
1		cbvmow.1	. 2 |- F/ y ph
2		cbvmow.2	. 2 |- F/ x ps
3		cbvmow.3	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvmo 2117	1 |- ( E* x ph <-> E* y ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1506   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:  cbvrmow  2714