Users' Mathboxes Mathbox for Giovanni Mascellani < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbali Structured version   Visualization version   GIF version

Theorem sbali 32884
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 2013 . . 3 𝑥𝑥𝜑
32sbcgf 3463 . 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 194  wal 1472  wcel 1975  Vcvv 3168  [wsbc 3397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-v 3170  df-sbc 3398
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator