Theorem sbcni 34223

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 3674	. . 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 325	1 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 197   ∈ wcel 2156  Vcvv 3391  [wsbc 3633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-12 2214  ax-13 2420  ax-ext 2784

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-v 3393  df-sbc 3634

This theorem is referenced by: (None)