Theorem sbcimdvOLD

Description: Obsolete version of sbcimdv as of 12-Oct-2024. (Contributed by NM, 11-Nov-2005) (Revised by NM, 17-Aug-2018) (Proof shortened by JJ, 7-Jul-2021) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sbcimdvOLD.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
Assertion	sbcimdvOLD	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbcimdvOLD.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		sbcex	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to A\in \mathrm{V}$$
3	1	alrimiv	$$⊢\phi \to \forall x\left(\psi \to \chi \right)$$
4		spsbc	$$⊢A\in \mathrm{V}\to \left(\forall x\left(\psi \to \chi \right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)$$
5		sbcim1	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$
6	3 4 5	syl56	$$⊢A\in \mathrm{V}\to \left(\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$
7	6	com3l	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \left(A\in \mathrm{V}\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$
8	2 7	mpdi	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$