Theorem 2sbcrexOLD

Description: Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014) Obsolete as of 24-Aug-2018. Use csbov123 instead. (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	2sbcrex.1	|- A e. _V
	2sbcrex.2	|- B e. _V
Assertion	2sbcrexOLD	|- ( [. A / a ]. [. B / b ]. E. c e. C ph <-> E. c e. C [. A / a ]. [. B / b ]. ph )

Proof

Step	Hyp	Ref	Expression
1		2sbcrex.1	|- A e. _V
2		2sbcrex.2	|- B e. _V
3		sbcrexgOLD	|- ( B e. _V -> ( [. B / b ]. E. c e. C ph <-> E. c e. C [. B / b ]. ph ) )
4	2 3	ax-mp	|- ( [. B / b ]. E. c e. C ph <-> E. c e. C [. B / b ]. ph )
5	4	sbcbii	|- ( [. A / a ]. [. B / b ]. E. c e. C ph <-> [. A / a ]. E. c e. C [. B / b ]. ph )
6		sbcrexgOLD	|- ( A e. _V -> ( [. A / a ]. E. c e. C [. B / b ]. ph <-> E. c e. C [. A / a ]. [. B / b ]. ph ) )
7	1 6	ax-mp	|- ( [. A / a ]. E. c e. C [. B / b ]. ph <-> E. c e. C [. A / a ]. [. B / b ]. ph )
8	5 7	bitri	|- ( [. A / a ]. [. B / b ]. E. c e. C ph <-> E. c e. C [. A / a ]. [. B / b ]. ph )