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Theorem nfexd2 1973
 Description: Variation on nfexd 1854 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 1542 . 2 (yψ ↔ ¬ y ¬ ψ)
2 nfald2.1 . . . 4 yφ
3 nfald2.2 . . . . 5 ((φ ¬ x x = y) → Ⅎxψ)
43nfnd 1791 . . . 4 ((φ ¬ x x = y) → Ⅎx ¬ ψ)
52, 4nfald2 1972 . . 3 (φ → Ⅎxy ¬ ψ)
65nfnd 1791 . 2 (φ → Ⅎx ¬ y ¬ ψ)
71, 6nfxfrd 1571 1 (φ → Ⅎxyψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  nfeud2  2216  nfmod2  2217
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