Theorem dvelimf 1997

Description: Version of dvelimv 1939 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
dvelimf.1	⊢ Ⅎxφ
dvelimf.2	⊢ Ⅎzψ
dvelimf.3	⊢ (z = y → (φ ↔ ψ))

Assertion

Ref	Expression
dvelimf	⊢ (¬ ∀x x = y → Ⅎxψ)

Proof of Theorem dvelimf

Step	Hyp	Ref	Expression
1		dvelimf.2	. . . 4 ⊢ Ⅎzψ
2		dvelimf.3	. . . 4 ⊢ (z = y → (φ ↔ ψ))
3	1, 2	equsal 1960	. . 3 ⊢ (∀z(z = y → φ) ↔ ψ)
4	3	bicomi 193	. 2 ⊢ (ψ ↔ ∀z(z = y → φ))
5		nfnae 1956	. . 3 ⊢ Ⅎz ¬ ∀x x = y
6		nfnae 1956	. . . . . 6 ⊢ Ⅎx ¬ ∀x x = y
7		nfnae 1956	. . . . . 6 ⊢ Ⅎx ¬ ∀x x = z
8	6, 7	nfan 1824	. . . . 5 ⊢ Ⅎx(¬ ∀x x = y ∧ ¬ ∀x x = z)
9		ax12o 1934	. . . . . 6 ⊢ (¬ ∀x x = z → (¬ ∀x x = y → (z = y → ∀x z = y)))
10	9	impcom 419	. . . . 5 ⊢ ((¬ ∀x x = y ∧ ¬ ∀x x = z) → (z = y → ∀x z = y))
11	8, 10	nfd 1766	. . . 4 ⊢ ((¬ ∀x x = y ∧ ¬ ∀x x = z) → Ⅎx z = y)
12		dvelimf.1	. . . . 5 ⊢ Ⅎxφ
13	12	a1i 10	. . . 4 ⊢ ((¬ ∀x x = y ∧ ¬ ∀x x = z) → Ⅎxφ)
14	11, 13	nfimd 1808	. . 3 ⊢ ((¬ ∀x x = y ∧ ¬ ∀x x = z) → Ⅎx(z = y → φ))
15	5, 14	nfald2 1972	. 2 ⊢ (¬ ∀x x = y → Ⅎx∀z(z = y → φ))
16	4, 15	nfxfrd 1571	1 ⊢ (¬ ∀x x = y → Ⅎxψ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  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:  dvelimnf  2017