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Theorem nfald2 1972
 Description: Variation on nfald 1852 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
nfald2 (φ → Ⅎxyψ)

Proof of Theorem nfald2
StepHypRef Expression
1 nfald2.1 . . . . 5 yφ
2 nfnae 1956 . . . . 5 y ¬ x x = y
31, 2nfan 1824 . . . 4 y(φ ¬ x x = y)
4 nfald2.2 . . . 4 ((φ ¬ x x = y) → Ⅎxψ)
53, 4nfald 1852 . . 3 ((φ ¬ x x = y) → Ⅎxyψ)
65ex 423 . 2 (φ → (¬ x x = y → Ⅎxyψ))
7 nfa1 1788 . . 3 yyψ
8 biidd 228 . . . 4 (x x = y → (yψyψ))
98drnf1 1969 . . 3 (x x = y → (Ⅎxyψ ↔ Ⅎyyψ))
107, 9mpbiri 224 . 2 (x x = y → Ⅎxyψ)
116, 10pm2.61d2 152 1 (φ → Ⅎxyψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  nfexd2  1973  dvelimf  1997  nfeud2  2216  nfrald  2665  nfiotad  4342
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