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Theorem sbali 35432
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 2156 . 2 𝑥𝑥𝜑
31, 2sbcgfi 3823 1 ([𝐴 / 𝑥]𝑥𝜑 ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536  wcel 2115  Vcvv 3471  [wsbc 3749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-sbc 3750
This theorem is referenced by: (None)
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