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Theorem sbali 33586
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 2025 . . 3 𝑥𝑥𝜑
32sbcgf 3488 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥𝜑 ↔ ∀𝑥𝜑))
41, 3ax-mp 5 1 ([𝐴 / 𝑥]𝑥𝜑 ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1478  wcel 1987  Vcvv 3190  [wsbc 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3192  df-sbc 3423
This theorem is referenced by: (None)
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