Theorem sbcni 33084

Description: Move class substitution inside a negation, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)

Hypotheses

Ref	Expression
sbcni.1	⊢ 𝐴 ∈ V
sbcni.2	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)

Assertion

Ref	Expression
sbcni	⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓)

Proof of Theorem sbcni

Step	Hyp	Ref	Expression
1		sbcni.1	. . 3 ⊢ 𝐴 ∈ V
2		sbcng 3443	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
3	1, 2	ax-mp 5	. 2 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)
4		sbcni.2	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)
5	3, 4	xchbinx 323	1 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 195   ∈ wcel 1977  Vcvv 3173  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403

This theorem is referenced by: (None)