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Theorem ceqsalg 2647
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
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 elisset 2633 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 nfa1 1479 . . . 4 𝑥𝑥(𝑥 = 𝐴𝜑)
3 ceqsalg.1 . . . 4 𝑥𝜓
4 ceqsalg.2 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
54biimpd 142 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
65a2i 11 . . . . 5 ((𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓))
76sps 1475 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓))
82, 3, 7exlimd 1533 . . 3 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴𝜓))
91, 8syl5com 29 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜓))
104biimprcd 158 . . 3 (𝜓 → (𝑥 = 𝐴𝜑))
113, 10alrimi 1460 . 2 (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))
129, 11impbid1 140 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287   = wceq 1289  wnf 1394  wex 1426  wcel 1438
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:  ceqsal  2648  sbc6g  2864  uniiunlem  3109  sucprcreg  4363  funimass4  5349  ralrnmpt2  5751
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