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Theorem nfald2 2444
Description: Variation on nfald 2322 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2371. Use nfald 2322 instead. (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 2433 . . . . 5 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
31, 2nfan 1903 . . . 4 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
4 nfald2.2 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
53, 4nfald 2322 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦𝜓)
65ex 414 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦𝜓))
7 nfa1 2149 . . 3 𝑦𝑦𝜓
8 biidd 262 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 ↔ ∀𝑦𝜓))
98drnf1 2442 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦𝜓 ↔ Ⅎ𝑦𝑦𝜓))
107, 9mpbiri 258 . 2 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦𝜓)
116, 10pm2.61d2 181 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wal 1540  wnf 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787
This theorem is referenced by:  nfexd2  2445  dvelimf  2447  nfmod2  2553  nfrald  3344  nfiotad  6454  nfixp  8858
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