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Theorem ceqsal 2789
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 2788 . 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 105  wal 1362   = wceq 1364  wnf 1471  wcel 2164  Vcvv 2760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762
This theorem is referenced by:  ceqsalv  2790
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