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Theorem 2exsb 1882
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 1881 . . . 4 (yφwy(y = wφ))
21exbii 1493 . . 3 (xyφxwy(y = wφ))
3 excom 1551 . . 3 (xwy(y = wφ) ↔ wxy(y = wφ))
42, 3bitri 173 . 2 (xyφwxy(y = wφ))
5 exsb 1881 . . . 4 (xy(y = wφ) ↔ zx(x = zy(y = wφ)))
6 impexp 250 . . . . . . . 8 (((x = z y = w) → φ) ↔ (x = z → (y = wφ)))
76albii 1356 . . . . . . 7 (y((x = z y = w) → φ) ↔ y(x = z → (y = wφ)))
8 19.21v 1750 . . . . . . 7 (y(x = z → (y = wφ)) ↔ (x = zy(y = wφ)))
97, 8bitr2i 174 . . . . . 6 ((x = zy(y = wφ)) ↔ y((x = z y = w) → φ))
109albii 1356 . . . . 5 (x(x = zy(y = wφ)) ↔ xy((x = z y = w) → φ))
1110exbii 1493 . . . 4 (zx(x = zy(y = wφ)) ↔ zxy((x = z y = w) → φ))
125, 11bitri 173 . . 3 (xy(y = wφ) ↔ zxy((x = z y = w) → φ))
1312exbii 1493 . 2 (wxy(y = wφ) ↔ wzxy((x = z y = w) → φ))
14 excom 1551 . 2 (wzxy((x = z y = w) → φ) ↔ zwxy((x = z y = w) → φ))
154, 13, 143bitri 195 1 (xyφzwxy((x = z y = w) → φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378
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-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-sb 1643
This theorem is referenced by: (None)
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