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Theorem drex1 1676
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) (Revised by NM, 3-Feb-2015.)
Hypothesis
Ref Expression
drex1.1 (x x = y → (φψ))
Assertion
Ref Expression
drex1 (x x = y → (xφyψ))

Proof of Theorem drex1
StepHypRef Expression
1 hbae 1603 . . . 4 (x x = yxx x = y)
2 drex1.1 . . . . 5 (x x = y → (φψ))
3 ax-4 1397 . . . . . 6 (x x = yx = y)
43biantrurd 289 . . . . 5 (x x = y → (ψ ↔ (x = y ψ)))
52, 4bitr2d 178 . . . 4 (x x = y → ((x = y ψ) ↔ φ))
61, 5exbidh 1502 . . 3 (x x = y → (x(x = y ψ) ↔ xφ))
7 ax11e 1674 . . . 4 (x = y → (x(x = y ψ) → yψ))
87sps 1427 . . 3 (x x = y → (x(x = y ψ) → yψ))
96, 8sylbird 159 . 2 (x x = y → (xφyψ))
10 hbae 1603 . . . 4 (x x = yyx x = y)
11 equcomi 1589 . . . . . . 7 (x = yy = x)
1211sps 1427 . . . . . 6 (x x = yy = x)
1312biantrurd 289 . . . . 5 (x x = y → (φ ↔ (y = x φ)))
1413, 2bitr3d 179 . . . 4 (x x = y → ((y = x φ) ↔ ψ))
1510, 14exbidh 1502 . . 3 (x x = y → (y(y = x φ) ↔ yψ))
16 ax11e 1674 . . . . 5 (y = x → (y(y = x φ) → xφ))
1716sps 1427 . . . 4 (y y = x → (y(y = x φ) → xφ))
1817alequcoms 1406 . . 3 (x x = y → (y(y = x φ) → xφ))
1915, 18sylbird 159 . 2 (x x = y → (yψxφ))
209, 19impbid 120 1 (x x = y → (xφyψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  drsb1  1677  exdistrfor  1678  copsexg  3972
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