Theorem sbcim2g

Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg . sbcim2g is sbcim2gVD without virtual deductions and was automatically derived from sbcim2gVD using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sbcim2g	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sbcimg	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\right)$$
2	1	biimpd	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\right)$$
3		sbcimg	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$
4		imbi2	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
5	4	biimpcd	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
6	2 3 5	syl6ci	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
7		idd	$$⊢A\in V\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
8		biimpr	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)$$
9	3 7 8	ee13	$$⊢A\in V\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\right)$$
10	9 1	sylibrd	$$⊢A\in V\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\right)$$
11	6 10	impbid	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$