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Theorem ceqsalg 2740
 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 2726 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 nfa1 1521 . . . 4 𝑥𝑥(𝑥 = 𝐴𝜑)
3 ceqsalg.1 . . . 4 𝑥𝜓
4 ceqsalg.2 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
54biimpd 143 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
65a2i 11 . . . . 5 ((𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓))
76sps 1517 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓))
82, 3, 7exlimd 1577 . . 3 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴𝜓))
91, 8syl5com 29 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜓))
104biimprcd 159 . . 3 (𝜓 → (𝑥 = 𝐴𝜑))
113, 10alrimi 1502 . 2 (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))
129, 11impbid1 141 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1333   = wceq 1335  Ⅎwnf 1440  ∃wex 1472   ∈ wcel 2128 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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714 This theorem is referenced by:  ceqsal  2741  sbc6g  2961  uniiunlem  3216  sucprcreg  4506  funimass4  5516  ralrnmpo  5929
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