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Theorem bj-drex1v 33068
Description: Version of drex1 2491 with a dv condition, which does not require ax-13 2422. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-drex1v.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-drex1v (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-drex1v
StepHypRef Expression
1 bj-drex1v.1 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
21notbid 309 . . . 4 (∀𝑥 𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
32bj-dral1v 33067 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓))
43notbid 309 . 2 (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑦 ¬ 𝜓))
5 df-ex 1860 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
6 df-ex 1860 . 2 (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
74, 5, 63bitr4g 305 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wal 1635  wex 1859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-10 2186  ax-12 2215
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ex 1860  df-nf 1864
This theorem is referenced by: (None)
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