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Theorem sbc6 3660
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 3659 . 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 198  wal 1651   = wceq 1653  wcel 2157  Vcvv 3385  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-v 3387  df-sbc 3634
This theorem is referenced by:  intab  4697  2sbc6g  39393
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