Theorem sbcim1

Description: Distribution of class substitution over implication. One direction of sbcimg that holds for proper classes. (Contributed by NM, 17-Aug-2018) Avoid ax-10 , ax-12 . (Revised by SN, 26-Oct-2024)

	Ref	Expression
Assertion	sbcim1	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbcex	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\to A\in \mathrm{V}$$
2		dfsbcq2	$$⊢y=A\to \left(\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\right)$$
3		dfsbcq2	$$⊢y=A\to \left(\left[y/x\right]\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
4		dfsbcq2	$$⊢y=A\to \left(\left[y/x\right]\psi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)$$
5	3 4	imbi12d	$$⊢y=A\to \left(\left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$
6	2 5	imbi12d	$$⊢y=A\to \left(\left(\left[y/x\right]\left(\phi \to \psi \right)\to \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)\right)$$
7		sbi1	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\to \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)$$
8	6 7	vtoclg	$$⊢A\in \mathrm{V}\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$
9	1 8	mpcom	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)$$