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

Theorem sbtru 32877
Description: Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.)
Hypothesis
Ref Expression
sbtru.1 𝐴 ∈ V
Assertion
Ref Expression
sbtru ([𝐴 / 𝑥]⊤ ↔ ⊤)

Proof of Theorem sbtru
StepHypRef Expression
1 sbtru.1 . 2 𝐴 ∈ V
2 sbcg 3465 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]⊤ ↔ ⊤))
31, 2ax-mp 5 1 ([𝐴 / 𝑥]⊤ ↔ ⊤)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wtru 1475  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-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