ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  drex1 GIF version

Theorem drex1 1726
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 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Proof of Theorem drex1
StepHypRef Expression
1 hbae 1653 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥 𝑥 = 𝑦)
2 drex1.1 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
3 ax-4 1445 . . . . . 6 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
43biantrurd 299 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜓 ↔ (𝑥 = 𝑦𝜓)))
52, 4bitr2d 187 . . . 4 (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦𝜓) ↔ 𝜑))
61, 5exbidh 1550 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) ↔ ∃𝑥𝜑))
7 ax11e 1724 . . . 4 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑦𝜓))
87sps 1475 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑦𝜓))
96, 8sylbird 168 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 → ∃𝑦𝜓))
10 hbae 1653 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)
11 equcomi 1637 . . . . . . 7 (𝑥 = 𝑦𝑦 = 𝑥)
1211sps 1475 . . . . . 6 (∀𝑥 𝑥 = 𝑦𝑦 = 𝑥)
1312biantrurd 299 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ (𝑦 = 𝑥𝜑)))
1413, 2bitr3d 188 . . . 4 (∀𝑥 𝑥 = 𝑦 → ((𝑦 = 𝑥𝜑) ↔ 𝜓))
1510, 14exbidh 1550 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝑦 = 𝑥𝜑) ↔ ∃𝑦𝜓))
16 ax11e 1724 . . . . 5 (𝑦 = 𝑥 → (∃𝑦(𝑦 = 𝑥𝜑) → ∃𝑥𝜑))
1716sps 1475 . . . 4 (∀𝑦 𝑦 = 𝑥 → (∃𝑦(𝑦 = 𝑥𝜑) → ∃𝑥𝜑))
1817alequcoms 1454 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝑦 = 𝑥𝜑) → ∃𝑥𝜑))
1915, 18sylbird 168 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑦𝜓 → ∃𝑥𝜑))
209, 19impbid 127 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  drsb1  1727  exdistrfor  1728  copsexg  4062
  Copyright terms: Public domain W3C validator