Theorem sbcn1 3827

Description: Move negation in and out of class substitution. One direction of sbcng 3822 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Assertion

Ref	Expression
sbcn1	⊢ ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)

Proof of Theorem sbcn1

Step	Hyp	Ref	Expression
1		sbcex 3785	. 2 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 → 𝐴 ∈ V)
2		sbcng 3822	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
3	2	biimpd 231	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑))
4	1, 3	mpcom 38	1 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2113  Vcvv 3497  [wsbc 3775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3499  df-sbc 3776

This theorem is referenced by: (None)