Theorem 2sbcrex

Description: Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014) (Revised by NM, 24-Aug-2018)

	Ref	Expression
Assertion	2sbcrex	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\exists c\in C\phi \leftrightarrow \exists c\in C\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		sbcrex	$$⊢\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\exists c\in C\phi \leftrightarrow \exists c\in C\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi$$
2	1	sbcbii	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\exists c\in C\phi \leftrightarrow \underset{\dot{}}{[}A/a\underset{\dot{}}{]}\exists c\in C\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi$$
3		sbcrex	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\exists c\in C\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi \leftrightarrow \exists c\in C\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi$$
4	2 3	bitri	$$⊢\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\exists c\in C\phi \leftrightarrow \exists c\in C\underset{\dot{}}{[}A/a\underset{\dot{}}{]}\underset{\dot{}}{[}B/b\underset{\dot{}}{]}\phi$$