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Theorem ceqsex2v 2641
 Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
Hypotheses
Ref Expression
ceqsex2v.1 𝐴 ∈ V
ceqsex2v.2 𝐵 ∈ V
ceqsex2v.3 (𝑥 = 𝐴 → (𝜑𝜓))
ceqsex2v.4 (𝑦 = 𝐵 → (𝜓𝜒))
Assertion
Ref Expression
ceqsex2v (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵𝜑) ↔ 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)

Proof of Theorem ceqsex2v
StepHypRef Expression
1 nfv 1462 . 2 𝑥𝜓
2 nfv 1462 . 2 𝑦𝜒
3 ceqsex2v.1 . 2 𝐴 ∈ V
4 ceqsex2v.2 . 2 𝐵 ∈ V
5 ceqsex2v.3 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
6 ceqsex2v.4 . 2 (𝑦 = 𝐵 → (𝜓𝜒))
71, 2, 3, 4, 5, 6ceqsex2 2640 1 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵𝜑) ↔ 𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103   ∧ w3a 920   = wceq 1285  ∃wex 1422   ∈ wcel 1434  Vcvv 2602 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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604 This theorem is referenced by:  ceqsex3v  2642  ceqsex4v  2643
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