Theorem sbcbid

Description: Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014)

	Ref	Expression
Hypotheses	sbcbid.1	$$⊢Ⅎx\phi$$
	sbcbid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	sbcbid	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbcbid.1	$$⊢Ⅎx\phi$$
2		sbcbid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	1 2	abbid	$$⊢\phi \to \left\{x|\psi \right\}=\left\{x|\chi \right\}$$
4	3	eleq2d	$$⊢\phi \to \left(A\in \left\{x|\psi \right\}\leftrightarrow A\in \left\{x|\chi \right\}\right)$$
5		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow A\in \left\{x|\psi \right\}$$
6		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \leftrightarrow A\in \left\{x|\chi \right\}$$
7	4 5 6	3bitr4g	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$