Theorem sbcim1 3775

Description: Distribution of class substitution over implication. One direction of sbcimg 3770 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) Avoid ax-10 2140, ax-12 2174. (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 3729	. 2 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → 𝐴 ∈ V)
2		dfsbcq2 3722	. . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ [𝐴 / 𝑥](𝜑 → 𝜓)))
3		dfsbcq2 3722	. . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
4		dfsbcq2 3722	. . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓))
5	3, 4	imbi12d 344	. . . 4 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
6	2, 5	imbi12d 344	. . 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 2077	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
8	6, 7	vtoclg 3503	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
9	1, 8	mpcom 38	1 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))

Syntax hints:   → wi 4   = wceq 1541  [wsb 2070   ∈ wcel 2109  Vcvv 3430  [wsbc 3719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-v 3432  df-sbc 3720

This theorem is referenced by:  sbcimdvOLD  3795  opsbc2ie  30803  frege59c  41483  frege60c  41484  frege62c  41486  frege65c  41489  frege70  41494  frege72  41496  frege92  41516  frege120  41544