Theorem sbali 38616

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 2187	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
3	1, 2	sbcgfi 3819	1 ⊢ ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)

Syntax hints:   ↔ wb 208  ∀wal 1560   ∈ wcel 2144  Vcvv 3456  [wsbc 3746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-sbc 3747

This theorem is referenced by: (None)