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Theorem ceqsalv 2716
 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
ceqsalv.1 𝐴 ∈ V
ceqsalv.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsalv (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsalv
StepHypRef Expression
1 nfv 1508 . 2 𝑥𝜓
2 ceqsalv.1 . 2 𝐴 ∈ V
3 ceqsalv.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3ceqsal 2715 1 (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329   = wceq 1331   ∈ wcel 1480  Vcvv 2686 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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688 This theorem is referenced by:  gencbval  2734  clel2  2818  clel4  2821  reu8  2880  raliunxp  4680  fv3  5444
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