Theorem sbcbi1

Description: Distribution of class substitution over biconditional. One direction of sbcbig that holds for proper classes. (Contributed by NM, 17-Aug-2018)

	Ref	Expression
Assertion	sbcbi1	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \leftrightarrow \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbcex	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \leftrightarrow \psi \right)\to A\in \mathrm{V}$$
2		sbcbig	$$⊢A\in \mathrm{V}\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$
3	2	biimpd	$$⊢A\in \mathrm{V}\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \leftrightarrow \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$
4	1 3	mpcom	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \leftrightarrow \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)$$