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Theorem drex1 2358
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.)
Hypothesis
Ref Expression
dral1.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
drex1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Proof of Theorem drex1
StepHypRef Expression
1 dral1.1 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
21notbid 307 . . . 4 (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
32dral1 2356 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓))
43notbid 307 . 2 (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑦 ¬ 𝜓))
5 df-ex 1745 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
6 df-ex 1745 . 2 (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
74, 5, 63bitr4g 303 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  exdistrf  2364  drsb1  2405  eujustALT  2501  copsexg  4985  dfid3  5054  dropab1  38968  dropab2  38969  e2ebind  39096  e2ebindVD  39462  e2ebindALT  39479
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