Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfald2 Structured version   Visualization version   GIF version

Theorem nfald2 2362
 Description: Variation on nfald 2201 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 2351 . . . . 5 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
31, 2nfan 1868 . . . 4 𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
4 nfald2.2 . . . 4 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
53, 4nfald 2201 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦𝜓)
65ex 449 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦𝜓))
7 nfa1 2068 . . 3 𝑦𝑦𝜓
8 biidd 252 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 ↔ ∀𝑦𝜓))
98drnf1 2360 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦𝜓 ↔ Ⅎ𝑦𝑦𝜓))
107, 9mpbiri 248 . 2 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦𝜓)
116, 10pm2.61d2 172 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1521  Ⅎwnf 1748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750 This theorem is referenced by:  nfexd2  2363  dvelimf  2365  nfeud2  2510  nfrald  2973  nfiotad  5892  nfixp  7969
 Copyright terms: Public domain W3C validator