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Theorem nfald2 2456
 Description: Variation on nfald 2336 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. Usage of this theorem is discouraged because it depends on ax-13 2379. Check out nfald 2336 for a version requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfald2.1 𝑦𝜑
nfald2.2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfald2 (𝜑 → Ⅎ𝑥𝑦𝜓)

Proof of Theorem nfald2
StepHypRef Expression
1 nfald2.1 . . . . 5 𝑦𝜑
2 nfnae 2445 . . . . 5 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
31, 2nfan 1900 . . . 4 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
4 nfald2.2 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
53, 4nfald 2336 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦𝜓)
65ex 416 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦𝜓))
7 nfa1 2152 . . 3 𝑦𝑦𝜓
8 biidd 265 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 ↔ ∀𝑦𝜓))
98drnf1 2454 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦𝜓 ↔ Ⅎ𝑦𝑦𝜓))
107, 9mpbiri 261 . 2 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦𝜓)
116, 10pm2.61d2 184 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀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-11 2158  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786 This theorem is referenced by:  nfexd2  2457  dvelimf  2459  nfmod2  2576  nfrald  3152  nfiotad  6299  nfixp  8499
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