Theorem sbcimg

Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004)

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

Proof

Step	Hyp	Ref	Expression
1		dfsbcq2	$$⊢y=A\to \left(\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\right)$$
2		dfsbcq2	$$⊢y=A\to \left(\left[y/x\right]\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
3		dfsbcq2	$$⊢y=A\to \left(\left[y/x\right]\psi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)$$
4	2 3	imbi12d	$$⊢y=A\to \left(\left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$
5		sbim	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)$$
6	1 4 5	vtoclbg	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$