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Theorem ceqsex2v 3462
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 2013 . 2 𝑥𝜓
2 nfv 2013 . 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 3461 1 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵𝜑) ↔ 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1111   = wceq 1656  wex 1878  wcel 2164  Vcvv 3414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416
This theorem is referenced by:  ceqsex3v  3463  ceqsex4v  3464  ispos  17307  elfuns  32556  brimg  32578  brapply  32579  brsuccf  32582  brrestrict  32590  dfrdg4  32592  diblsmopel  37241
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