Theorem cbvrmov 3174

Description: Change the bound variable of a restricted uniqueness 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 1843	. 2 |- F/ y ph
2		nfv 1843	. 2 |- F/ x ps
3		cbvralv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvrmo 3170	1 |- ( E* x e. A ph <-> E* y e. A ps )

Syntax hints:   -> wi 4   <-> wb 196   E*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920

This theorem is referenced by: (None)