Theorem cbvmo

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvmow when possible. (Contributed by NM, 9-Mar-1995) (Revised by Andrew Salmon, 8-Jun-2011) (Proof shortened by Wolf Lammen, 4-Jan-2023) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvmo.1	|- F/ y ph
	cbvmo.2	|- F/ x ps
	cbvmo.3	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvmo	|- ( E* x ph <-> E* y ps )

Proof

Step	Hyp	Ref	Expression
1		cbvmo.1	|- F/ y ph
2		cbvmo.2	|- F/ x ps
3		cbvmo.3	|- ( x = y -> ( ph <-> ps ) )
4	1	sb8mo	|- ( E* x ph <-> E* y [ y / x ] ph )
5	2 3	sbie	|- ( [ y / x ] ph <-> ps )
6	5	mobii	|- ( E* y [ y / x ] ph <-> E* y ps )
7	4 6	bitri	|- ( E* x ph <-> E* y ps )