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Theorem nfexd2 2470
Description: Variation on nfexd 2350 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 2392. Check out nfexd 2350 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
nfexd2 (𝜑 → Ⅎ𝑥𝑦𝜓)

Proof of Theorem nfexd2
StepHypRef Expression
1 df-ex 1782 . 2 (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
2 nfald2.1 . . . 4 𝑦𝜑
3 nfald2.2 . . . . 5 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
43nfnd 1859 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 ¬ 𝜓)
52, 4nfald2 2469 . . 3 (𝜑 → Ⅎ𝑥𝑦 ¬ 𝜓)
65nfnd 1859 . 2 (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓)
71, 6nfxfrd 1855 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1536  wex 1781  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-11 2162  ax-12 2179  ax-13 2392
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:  nfeud2  2677
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