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Theorem drnf1 2454
 Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker drnf1v 2378 if possible. (Contributed by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
dral1.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
drnf1 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Proof of Theorem drnf1
StepHypRef Expression
1 dral1.1 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
21dral1 2450 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
31, 2imbi12d 348 . . 3 (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑦𝜓)))
43dral1 2450 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜓 → ∀𝑦𝜓)))
5 nf5 2286 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
6 nf5 2286 . 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  Ⅎwnf 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 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379 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:  nfald2  2456  drnfc1  2974  wl-nfs1t  35093
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