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Theorem ceqsex2 2588
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 888 . . . . 5 ((x = A y = B φ) ↔ (x = A (y = B φ)))
21exbii 1493 . . . 4 (y(x = A y = B φ) ↔ y(x = A (y = B φ)))
3 19.42v 1783 . . . 4 (y(x = A (y = B φ)) ↔ (x = A y(y = B φ)))
42, 3bitri 173 . . 3 (y(x = A y = B φ) ↔ (x = A y(y = B φ)))
54exbii 1493 . 2 (xy(x = A y = B φ) ↔ x(x = A y(y = B φ)))
6 nfv 1418 . . . . 5 x y = B
7 ceqsex2.1 . . . . 5 xψ
86, 7nfan 1454 . . . 4 x(y = B ψ)
98nfex 1525 . . 3 xy(y = B ψ)
10 ceqsex2.3 . . 3 A V
11 ceqsex2.5 . . . . 5 (x = A → (φψ))
1211anbi2d 437 . . . 4 (x = A → ((y = B φ) ↔ (y = B ψ)))
1312exbidv 1703 . . 3 (x = A → (y(y = B φ) ↔ y(y = B ψ)))
149, 10, 13ceqsex 2586 . 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 2586 . 2 (y(y = B ψ) ↔ χ)
195, 14, 183bitri 195 1 (xy(x = A y = B φ) ↔ χ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242  wnf 1346  wex 1378   wcel 1390  Vcvv 2551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  ceqsex2v  2589
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