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Theorem drex1v 2390
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drex1 2465 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by NM, 27-Feb-2005.) (Revised by BJ, 17-Jun-2019.)
Hypothesis
Ref Expression
dral1v.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
drex1v (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem drex1v
StepHypRef Expression
1 dral1v.1 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
21notbid 321 . . . 4 (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
32dral1v 2389 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓))
43notbid 321 . 2 (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑦 ¬ 𝜓))
5 df-ex 1782 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
6 df-ex 1782 . 2 (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
74, 5, 63bitr4g 317 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by:  copsexgw  5368  oprabidw  7180
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