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	$$⊢A\in \mathrm{V}$$
	2sbcrex.2	$$⊢B\in \mathrm{V}$$
Assertion	2sbcrexOLD	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\exists c\in C\phi \leftrightarrow \exists c\in C\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		2sbcrex.1	$$⊢A\in \mathrm{V}$$
2		2sbcrex.2	$$⊢B\in \mathrm{V}$$
3		sbcrexgOLD	$$⊢B\in \mathrm{V}\to \left(\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\exists c\in C\phi \leftrightarrow \exists c\in C\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi \right)$$
4	2 3	ax-mp	$$⊢\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\exists c\in C\phi \leftrightarrow \exists c\in C\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi$$
5	4	sbcbii	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\exists c\in C\phi \leftrightarrow \underset{\dot{}}{[}A/a\underset{\dot{}}{]}\exists c\in C\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi$$
6		sbcrexgOLD	$$⊢A\in \mathrm{V}\to \left(\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\exists c\in C\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi \leftrightarrow \exists c\in C\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi \right)$$
7	1 6	ax-mp	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\exists c\in C\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi \leftrightarrow \exists c\in C\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi$$
8	5 7	bitri	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\exists c\in C\phi \leftrightarrow \exists c\in C\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi$$