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Theorem drex1 2462
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 310 . . . 4 (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
32dral1 2460 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓))
43notbid 310 . 2 (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑦 ¬ 𝜓))
5 df-ex 1879 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
6 df-ex 1879 . 2 (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
74, 5, 63bitr4g 306 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wal 1654  wex 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-12 2220  ax-13 2389
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883
This theorem is referenced by:  exdistrf  2468  drsb1  2508  eujustALT  2643  copsexg  5178  dfid3  5253  dropab1  39488  dropab2  39489  e2ebind  39606  e2ebindVD  39965  e2ebindALT  39982
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