Theorem cbvrmov 2704

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypothesis

Ref	Expression
cbvralv.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvrmov	|- ( E* x e. A ph <-> E* y e. A ps )

Distinct variable groups:   x, A   y, A   ph, y   ps, x

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

Proof of Theorem cbvrmov

Step	Hyp	Ref	Expression
1		nfv 1526	. 2 |- F/ y ph
2		nfv 1526	. 2 |- F/ x ps
3		cbvralv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvrmo 2700	1 |- ( E* x e. A ph <-> E* y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   E*wrmo 2456

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-reu 2460  df-rmo 2461

This theorem is referenced by: (None)