Theorem sbcim1 3710

Description: Distribution of class substitution over implication. One direction of sbcimg 3705 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Assertion

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

Proof of Theorem sbcim1

Step	Hyp	Ref	Expression
1		sbcex 3673	. 2 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → 𝐴 ∈ V)
2		sbcimg 3705	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
3	2	biimpd 221	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
4	1, 3	mpcom 38	1 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))

Syntax hints:   → wi 4   ∈ wcel 2166  Vcvv 3415  [wsbc 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-12 2222  ax-13 2391  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-v 3417  df-sbc 3664

This theorem is referenced by:  sbcimdv  3725  frege59c  39057  frege60c  39058  frege62c  39060  frege65c  39063  frege70  39068  frege72  39070  frege92  39090  frege120  39118