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Theorem 2exsb 2132
 Description: An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
Assertion
Ref Expression
2exsb (xyφzwxy((x = z y = w) → φ))
Distinct variable groups:   x,y,z   y,w,z   φ,z,w
Allowed substitution hints:   φ(x,y)

Proof of Theorem 2exsb
StepHypRef Expression
1 exsb 2130 . . . 4 (yφwy(y = wφ))
21exbii 1582 . . 3 (xyφxwy(y = wφ))
3 excom 1741 . . 3 (xwy(y = wφ) ↔ wxy(y = wφ))
42, 3bitri 240 . 2 (xyφwxy(y = wφ))
5 exsb 2130 . . . . 5 (xy(y = wφ) ↔ zx(x = zy(y = wφ)))
6 impexp 433 . . . . . . . . 9 (((x = z y = w) → φ) ↔ (x = z → (y = wφ)))
76albii 1566 . . . . . . . 8 (y((x = z y = w) → φ) ↔ y(x = z → (y = wφ)))
8 19.21v 1890 . . . . . . . 8 (y(x = z → (y = wφ)) ↔ (x = zy(y = wφ)))
97, 8bitr2i 241 . . . . . . 7 ((x = zy(y = wφ)) ↔ y((x = z y = w) → φ))
109albii 1566 . . . . . 6 (x(x = zy(y = wφ)) ↔ xy((x = z y = w) → φ))
1110exbii 1582 . . . . 5 (zx(x = zy(y = wφ)) ↔ zxy((x = z y = w) → φ))
125, 11bitri 240 . . . 4 (xy(y = wφ) ↔ zxy((x = z y = w) → φ))
1312exbii 1582 . . 3 (wxy(y = wφ) ↔ wzxy((x = z y = w) → φ))
14 excom 1741 . . 3 (wzxy((x = z y = w) → φ) ↔ zwxy((x = z y = w) → φ))
1513, 14bitri 240 . 2 (wxy(y = wφ) ↔ zwxy((x = z y = w) → φ))
164, 15bitri 240 1 (xyφzwxy((x = z y = w) → φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541 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-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  2eu6  2289
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