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	⊢ 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 1956	. . . . 5 ⊢ Ⅎy ¬ ∀x x = y
3	1, 2	nfan 1824	. . . 4 ⊢ Ⅎy(φ ∧ ¬ ∀x x = y)
4		nfald2.2	. . . 4 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)
5	3, 4	nfald 1852	. . 3 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx∀yψ)
6	5	ex 423	. 2 ⊢ (φ → (¬ ∀x x = y → Ⅎx∀yψ))
7		nfa1 1788	. . 3 ⊢ Ⅎy∀yψ
8		biidd 228	. . . 4 ⊢ (∀x x = y → (∀yψ ↔ ∀yψ))
9	8	drnf1 1969	. . 3 ⊢ (∀x x = y → (Ⅎx∀yψ ↔ Ⅎy∀yψ))
10	7, 9	mpbiri 224	. 2 ⊢ (∀x x = y → Ⅎx∀yψ)
11	6, 10	pm2.61d2 152	1 ⊢ (φ → Ⅎx∀yψ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is used by:  nfexd2  1973  dvelimf  1997  nfeud2  2216  nfrald  2666  nfiotad  4343