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Theorem drnf1v 2391
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2467 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.)
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))
21dral1v 2389 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
31, 2imbi12d 348 . . 3 (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑦𝜓)))
43dral1v 2389 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜓 → ∀𝑦𝜓)))
5 nf5 2292 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
6 nf5 2292 . 2 (Ⅎ𝑦𝜓 ↔ ∀𝑦(𝜓 → ∀𝑦𝜓))
74, 5, 63bitr4g 317 1 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wnf 1785
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:  nfriotadw  7115
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