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Theorem ceqsal 2648
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 2647 . 2 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 4ax-mp 7 1 (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287   = wceq 1289  wnf 1394  wcel 1438  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  ceqsalv  2649
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