Theorem sbcim1OLD

Description: Obsolete version of sbcim1 as of 26-Oct-2024. (Contributed by NM, 17-Aug-2018) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sbcim1OLD	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)$$

Proof

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