Theorem sbcim1 3791

Description: Distribution of class substitution over implication. One direction of sbcimg 3786 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) Avoid ax-10 2177, ax-12 2212. (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 3748	. 2 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → 𝐴 ∈ V)
2		dfsbcq2 3741	. . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ [𝐴 / 𝑥](𝜑 → 𝜓)))
3		dfsbcq2 3741	. . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
4		dfsbcq2 3741	. . . . 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 2107	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
8	6, 7	vtoclg 3517	. 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 1569  [wsb 2098   ∈ wcel 2144  Vcvv 3450  [wsbc 3738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-sbc 3739

This theorem is used by:  opsbc2ie  33120  frege59c  45016  frege60c  45017  frege62c  45019  frege65c  45022  frege70  45027  frege72  45029  frege92  45049  frege120  45077