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Theorem sbali 38119
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
StepHypRef Expression
1 sbali.1 . 2 𝐴 ∈ V
2 nfa1 2151 . 2 𝑥𝑥𝜑
31, 2sbcgfi 3864 1 ([𝐴 / 𝑥]𝑥𝜑 ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1538  wcel 2108  Vcvv 3480  [wsbc 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3789
This theorem is referenced by: (None)
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