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Theorem nfexd2 2412
Description: Variation on nfexd 2305 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
nfald2.1 𝑦𝜑
nfald2.2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfexd2 (𝜑 → Ⅎ𝑥𝑦𝜓)

Proof of Theorem nfexd2
StepHypRef Expression
1 df-ex 1824 . 2 (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
2 nfald2.1 . . . 4 𝑦𝜑
3 nfald2.2 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
43nfnd 1903 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 ¬ 𝜓)
52, 4nfald2 2411 . . 3 (𝜑 → Ⅎ𝑥𝑦 ¬ 𝜓)
65nfnd 1903 . 2 (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓)
71, 6nfxfrd 1898 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wal 1599  wex 1823  wnf 1827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828
This theorem is referenced by:  nfeud2  2611  nfmod2OLD  2624
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