Theorem sbc2rexgOLD

Description: Exchange a substitution with two existentials. (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
Assertion	sbc2rexgOLD	|- ( A e. V -> ( [. A / a ]. E. b e. B E. c e. C ph <-> E. b e. B E. c e. C [. A / a ]. ph ) )

Proof

Step	Hyp	Ref	Expression
1		elex	|- ( A e. V -> A e. _V )
2		sbcrexgOLD	|- ( A e. _V -> ( [. A / a ]. E. b e. B E. c e. C ph <-> E. b e. B [. A / a ]. E. c e. C ph ) )
3		sbcrexgOLD	|- ( A e. _V -> ( [. A / a ]. E. c e. C ph <-> E. c e. C [. A / a ]. ph ) )
4	3	rexbidv	|- ( A e. _V -> ( E. b e. B [. A / a ]. E. c e. C ph <-> E. b e. B E. c e. C [. A / a ]. ph ) )
5	2 4	bitrd	|- ( A e. _V -> ( [. A / a ]. E. b e. B E. c e. C ph <-> E. b e. B E. c e. C [. A / a ]. ph ) )
6	1 5	syl	|- ( A e. V -> ( [. A / a ]. E. b e. B E. c e. C ph <-> E. b e. B E. c e. C [. A / a ]. ph ) )