Theorem sbcim2gVD

Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg . The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcim2g is sbcim2gVD without virtual deductions and was automatically derived from sbcim2gVD .

	Ref	Expression
Assertion	sbcim2gVD	$$⊢A\in B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		idn1	$$⊢\left(A\in B\to A\in B\right)$$
2		idn2	$$⊢\left(A\in B,\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\right)$$
3		sbcimg	$$⊢A\in B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\right)$$
4	3	biimpd	$$⊢A\in B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\right)$$
5	1 2 4	e12	$$⊢\left(A\in B,\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\right)$$
6		sbcimg	$$⊢A\in B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$
7	1 6	e1a	$$⊢\left(A\in B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
8		imbi2	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
9	8	biimpcd	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
10	5 7 9	e21	$$⊢\left(A\in B,\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
11	10	in2	$$⊢\left(A\in B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)\right)$$
12		idn2	$$⊢\left(A\in B,\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
13		biimpr	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)$$
14	13	imim2d	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\right)$$
15	7 12 14	e12	$$⊢\left(A\in B,\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\right)$$
16	1 3	e1a	$$⊢\left(A\in B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\right)\right)$$
17		biimpr	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\right)\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\right)$$
18	17	com12	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)\right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\right)$$
19	15 16 18	e21	$$⊢\left(A\in B,\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\right)$$
20	19	in2	$$⊢\left(A\in B\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\right)\right)$$
21		impbi	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)\to \left(\left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)\right)$$
22	11 20 21	e11	$$⊢\left(A\in B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)\right)$$
23	22	in1	$$⊢A\in B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$