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Theorem ceqsex2v 2896
 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 = A y = B φ) ↔ 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 1619 . 2 xψ
2 nfv 1619 . 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 2895 1 (xy(x = A y = B φ) ↔ χ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  ceqsex3v  2897  ceqsex4v  2898  opksnelsik  4265  sikexlem  4295  br1stg  4730  elswap  4740  brswap2  4860  brsnsi  5773  oqelins4  5794  dmpprod  5840  lecex  6115  addccan2nclem1  6263  nmembers1lem1  6268
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