Theorem sbali 38711

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 2193	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
3	1, 2	sbcgfi 3825	1 ⊢ ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 209  ∀wal 1566   ∈ wcel 2150  Vcvv 3462  [wsbc 3752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-sbc 3753

This theorem is referenced by: (None)