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Theorem ceqsex2v 2701
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 1493 . 2 𝑥𝜓
2 nfv 1493 . 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 2700 1 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵𝜑) ↔ 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  w3a 947   = wceq 1316  wex 1453  wcel 1465  Vcvv 2660
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662
This theorem is referenced by:  ceqsex3v  2702  ceqsex4v  2703
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