Theorem sbali 33085

Description: Discard class substitution in a universal quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)

Hypothesis

Ref	Expression
sbali.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
sbali	⊢ ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)

Proof of Theorem sbali

Step	Hyp	Ref	Expression
1		sbali.1	. 2 ⊢ 𝐴 ∈ V
2		nfa1 2015	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	sbcgf 3468	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑))
4	1, 3	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 195  ∀wal 1473   ∈ 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)