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Theorem drnf1v 2361
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2434 with a disjoint variable condition, which does not require ax-13 2363. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) Avoid ax-10 2129. (Revised by Gino Giotto, 18-Nov-2024.)
Hypothesis
Ref Expression
dral1v.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
drnf1v (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem drnf1v
StepHypRef Expression
1 dral1v.1 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
21drex1v 2360 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
31dral1v 2358 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
42, 3imbi12d 344 . 2 (∀𝑥 𝑥 = 𝑦 → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑦𝜓 → ∀𝑦𝜓)))
5 df-nf 1778 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6 df-nf 1778 . 2 (Ⅎ𝑦𝜓 ↔ (∃𝑦𝜓 → ∀𝑦𝜓))
74, 5, 63bitr4g 314 1 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778
This theorem is referenced by:  nfriotadw  7366
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