Theorem sbcni 38493

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 3772	. . 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 336	1 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 208   ∈ wcel 2121  Vcvv 3433  [wsbc 3725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-sbc 3726

This theorem is referenced by: (None)