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Theorem ceqsex2v 3276
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 1883 . 2 𝑥𝜓
2 nfv 1883 . 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 3275 1 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵𝜑) ↔ 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233
This theorem is referenced by:  ceqsex3v  3277  ceqsex4v  3278  ispos  16994  elfuns  32147  brimg  32169  brapply  32170  brsuccf  32173  brrestrict  32181  dfrdg4  32183  diblsmopel  36777
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