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Theorem ceqsal 3448
 Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
Hypotheses
Ref Expression
ceqsal.1 𝑥𝜓
ceqsal.2 𝐴 ∈ V
ceqsal.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsal (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ceqsal
StepHypRef Expression
1 ceqsal.2 . 2 𝐴 ∈ V
2 ceqsal.1 . . 3 𝑥𝜓
3 ceqsal.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3ceqsalg 3446 . 2 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 4ax-mp 5 1 (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1537   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2112  Vcvv 3410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-clel 2831 This theorem is referenced by:  ceqsalv  3449  ralxp3f  33197  aomclem6  40369
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