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

Theorem spcgf 3261
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)
Hypotheses
Ref Expression
spcgf.1 𝑥𝐴
spcgf.2 𝑥𝜓
spcgf.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
spcgf (𝐴𝑉 → (∀𝑥𝜑𝜓))

Proof of Theorem spcgf
StepHypRef Expression
1 spcgf.2 . . 3 𝑥𝜓
2 spcgf.1 . . 3 𝑥𝐴
31, 2spcgft 3258 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉 → (∀𝑥𝜑𝜓)))
4 spcgf.3 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpg 1715 1 (𝐴𝑉 → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473   = wceq 1475  wnf 1699  wcel 1977  wnfc 2738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175
This theorem is referenced by:  spcegf  3262  spcgv  3266  rspc  3276  elabgt  3316  eusvnf  4787  gropd  25708  grstructd  25709
  Copyright terms: Public domain W3C validator