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Theorem ceqsal 2718
 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 2717 . 2 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 4ax-mp 5 1 (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330   = wceq 1332  Ⅎwnf 1437   ∈ wcel 1481  Vcvv 2689 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691 This theorem is referenced by:  ceqsalv  2719
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