MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsbc1 Structured version   Visualization version   GIF version

Theorem nfsbc1 3823
Description: Bound-variable hypothesis builder for class substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypothesis
Ref Expression
nfsbc1.1 𝑥𝐴
Assertion
Ref Expression
nfsbc1 𝑥[𝐴 / 𝑥]𝜑

Proof of Theorem nfsbc1
StepHypRef Expression
1 nfsbc1.1 . . . 4 𝑥𝐴
21a1i 11 . . 3 (⊤ → 𝑥𝐴)
32nfsbc1d 3822 . 2 (⊤ → Ⅎ𝑥[𝐴 / 𝑥]𝜑)
43mptru 1544 1 𝑥[𝐴 / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wtru 1538  wnf 1781  wnfc 2893  [wsbc 3804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-sbc 3805
This theorem is referenced by:  nfsbc1v  3824  riotass2  7435  riotass  7436  uzwo4  44955
  Copyright terms: Public domain W3C validator