Theorem sbali 38099

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 2149	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
3	1, 2	sbcgfi 3872	1 ⊢ ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1535   ∈ wcel 2106  Vcvv 3478  [wsbc 3791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792

This theorem is referenced by: (None)