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Theorem ceqsex2 2895
 Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
Hypotheses
Ref Expression
ceqsex2.1 xψ
ceqsex2.2 yχ
ceqsex2.3 A V
ceqsex2.4 B V
ceqsex2.5 (x = A → (φψ))
ceqsex2.6 (y = B → (ψχ))
Assertion
Ref Expression
ceqsex2 (xy(x = A y = B φ) ↔ χ)
Distinct variable groups:   x,y,A   x,B,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)   χ(x,y)

Proof of Theorem ceqsex2
StepHypRef Expression
1 3anass 938 . . . . 5 ((x = A y = B φ) ↔ (x = A (y = B φ)))
21exbii 1582 . . . 4 (y(x = A y = B φ) ↔ y(x = A (y = B φ)))
3 19.42v 1905 . . . 4 (y(x = A (y = B φ)) ↔ (x = A y(y = B φ)))
42, 3bitri 240 . . 3 (y(x = A y = B φ) ↔ (x = A y(y = B φ)))
54exbii 1582 . 2 (xy(x = A y = B φ) ↔ x(x = A y(y = B φ)))
6 nfv 1619 . . . . 5 x y = B
7 ceqsex2.1 . . . . 5 xψ
86, 7nfan 1824 . . . 4 x(y = B ψ)
98nfex 1843 . . 3 xy(y = B ψ)
10 ceqsex2.3 . . 3 A V
11 ceqsex2.5 . . . . 5 (x = A → (φψ))
1211anbi2d 684 . . . 4 (x = A → ((y = B φ) ↔ (y = B ψ)))
1312exbidv 1626 . . 3 (x = A → (y(y = B φ) ↔ y(y = B ψ)))
149, 10, 13ceqsex 2893 . 2 (x(x = A y(y = B φ)) ↔ y(y = B ψ))
15 ceqsex2.2 . . 3 yχ
16 ceqsex2.4 . . 3 B V
17 ceqsex2.6 . . 3 (y = B → (ψχ))
1815, 16, 17ceqsex 2893 . 2 (y(y = B ψ) ↔ χ)
195, 14, 183bitri 262 1 (xy(x = A y = B φ) ↔ χ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Vcvv 2859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  ceqsex2v  2896
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