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Theorem ceqsalg 3464
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3465. (Contributed by NM, 29-Oct-2003.) (Proof shortened by BJ, 29-Sep-2019.)
Hypotheses
Ref Expression
ceqsalg.1 𝑥𝜓
ceqsalg.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsalg (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem ceqsalg
StepHypRef Expression
1 ceqsalg.1 . 2 𝑥𝜓
2 ceqsalg.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32ax-gen 1798 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 ceqsalt 3462 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 3, 4mp3an12 1450 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537   = wceq 1539  wnf 1786  wcel 2106
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-clel 2816
This theorem is referenced by:  ceqsal  3466  clel2gOLD  3589  uniiunlem  4019  ralrnmpo  7412  fimaxre3  11921  pmapglbx  37783
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