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Theorem drex1 1771
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 1697 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥 𝑥 = 𝑦)
2 drex1.1 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
3 ax-4 1488 . . . . . 6 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
43biantrurd 303 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜓 ↔ (𝑥 = 𝑦𝜓)))
52, 4bitr2d 188 . . . 4 (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦𝜓) ↔ 𝜑))
61, 5exbidh 1594 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) ↔ ∃𝑥𝜑))
7 ax11e 1769 . . . 4 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑦𝜓))
87sps 1518 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜓) → ∃𝑦𝜓))
96, 8sylbird 169 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 → ∃𝑦𝜓))
10 hbae 1697 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)
11 equcomi 1681 . . . . . . 7 (𝑥 = 𝑦𝑦 = 𝑥)
1211sps 1518 . . . . . 6 (∀𝑥 𝑥 = 𝑦𝑦 = 𝑥)
1312biantrurd 303 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ (𝑦 = 𝑥𝜑)))
1413, 2bitr3d 189 . . . 4 (∀𝑥 𝑥 = 𝑦 → ((𝑦 = 𝑥𝜑) ↔ 𝜓))
1510, 14exbidh 1594 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝑦 = 𝑥𝜑) ↔ ∃𝑦𝜓))
16 ax11e 1769 . . . . 5 (𝑦 = 𝑥 → (∃𝑦(𝑦 = 𝑥𝜑) → ∃𝑥𝜑))
1716sps 1518 . . . 4 (∀𝑦 𝑦 = 𝑥 → (∃𝑦(𝑦 = 𝑥𝜑) → ∃𝑥𝜑))
1817alequcoms 1497 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝑦 = 𝑥𝜑) → ∃𝑥𝜑))
1915, 18sylbird 169 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑦𝜓 → ∃𝑥𝜑))
209, 19impbid 128 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1330   = wceq 1332  wex 1469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  drsb1  1772  exdistrfor  1773  copsexg  4172
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