Theorem nfald2 2450

Description: Variation on nfald 2329 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2377. Use nfald 2329 instead. (New usage is discouraged.)

Hypotheses

Ref	Expression
nfald2.1	⊢ Ⅎ𝑦𝜑
nfald2.2	⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfald2	⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)

Proof of Theorem nfald2

Step	Hyp	Ref	Expression
1		nfald2.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2		nfnae 2439	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
3	1, 2	nfan 1899	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
4		nfald2.2	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
5	3, 4	nfald 2329	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥∀𝑦𝜓)
6	5	ex 412	. 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦𝜓))
7		nfa1 2152	. . 3 ⊢ Ⅎ𝑦∀𝑦𝜓
8		biidd 262	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 ↔ ∀𝑦𝜓))
9	8	drnf1 2448	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥∀𝑦𝜓 ↔ Ⅎ𝑦∀𝑦𝜓))
10	7, 9	mpbiri 258	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦𝜓)
11	6, 10	pm2.61d2 181	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1538  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784

This theorem is referenced by:  nfexd2  2451  dvelimf  2453  nfmod2  2558  nfrald  3356  nfiotad  6494  nfixp  8936