MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbc6 Structured version   Visualization version   GIF version

Theorem sbc6 3429
Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
Hypothesis
Ref Expression
sbc6.1 𝐴 ∈ V
Assertion
Ref Expression
sbc6 ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sbc6
StepHypRef Expression
1 sbc6.1 . 2 𝐴 ∈ V
2 sbc6g 3428 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
31, 2ax-mp 5 1 ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473   = wceq 1475  wcel 1977  Vcvv 3173  [wsbc 3402
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-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-v 3175  df-sbc 3403
This theorem is referenced by:  intab  4442  2sbc6g  37638
  Copyright terms: Public domain W3C validator