Theorem sbcim1 3795

Description: Distribution of class substitution over implication. One direction of sbcimg 3790 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) Avoid ax-10 2180, ax-12 2216. (Revised by SN, 26-Oct-2024.)

Assertion

Ref	Expression
sbcim1	⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))

Proof of Theorem sbcim1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3752	. 2 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → 𝐴 ∈ V)
2		dfsbcq2 3745	. . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ [𝐴 / 𝑥](𝜑 → 𝜓)))
3		dfsbcq2 3745	. . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
4		dfsbcq2 3745	. . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓))
5	3, 4	imbi12d 347	. . . 4 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
6	2, 5	imbi12d 347	. . 3 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) ↔ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))))
7		sbi1 2110	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
8	6, 7	vtoclg 3521	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
9	1, 8	mpcom 39	1 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  [wsb 2101   ∈ wcel 2147  Vcvv 3454  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-sbc 3743

This theorem is used by:  opsbc2ie  33147  frege59c  45125  frege60c  45126  frege62c  45128  frege65c  45131  frege70  45136  frege72  45138  frege92  45158  frege120  45186