Theorem dvelimh 1964

Description: Version of dvelim 2016 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)

Hypotheses

Ref	Expression
dvelimh.1	⊢ (φ → ∀xφ)
dvelimh.2	⊢ (ψ → ∀zψ)
dvelimh.3	⊢ (z = y → (φ ↔ ψ))

Assertion

Ref	Expression
dvelimh	⊢ (¬ ∀x x = y → (ψ → ∀xψ))

Proof of Theorem dvelimh

Step	Hyp	Ref	Expression
1		hba1 1786	. . . . 5 ⊢ (∀z(z = y → φ) → ∀z∀z(z = y → φ))
2		ax10o 1952	. . . . . 6 ⊢ (∀z z = x → (∀z∀z(z = y → φ) → ∀x∀z(z = y → φ)))
3	2	aecoms 1947	. . . . 5 ⊢ (∀x x = z → (∀z∀z(z = y → φ) → ∀x∀z(z = y → φ)))
4	1, 3	syl5 28	. . . 4 ⊢ (∀x x = z → (∀z(z = y → φ) → ∀x∀z(z = y → φ)))
5	4	a1d 22	. . 3 ⊢ (∀x x = z → (¬ ∀x x = y → (∀z(z = y → φ) → ∀x∀z(z = y → φ))))
6		hbnae 1955	. . . . . 6 ⊢ (¬ ∀x x = z → ∀z ¬ ∀x x = z)
7		hbnae 1955	. . . . . 6 ⊢ (¬ ∀x x = y → ∀z ¬ ∀x x = y)
8	6, 7	hban 1828	. . . . 5 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → ∀z(¬ ∀x x = z ∧ ¬ ∀x x = y))
9		hbnae 1955	. . . . . . 7 ⊢ (¬ ∀x x = z → ∀x ¬ ∀x x = z)
10		hbnae 1955	. . . . . . 7 ⊢ (¬ ∀x x = y → ∀x ¬ ∀x x = y)
11	9, 10	hban 1828	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → ∀x(¬ ∀x x = z ∧ ¬ ∀x x = y))
12		ax12o 1934	. . . . . . 7 ⊢ (¬ ∀x x = z → (¬ ∀x x = y → (z = y → ∀x z = y)))
13	12	imp 418	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → (z = y → ∀x z = y))
14		dvelimh.1	. . . . . . 7 ⊢ (φ → ∀xφ)
15	14	a1i 10	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → (φ → ∀xφ))
16	11, 13, 15	hbimd 1815	. . . . 5 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → ((z = y → φ) → ∀x(z = y → φ)))
17	8, 16	hbald 1740	. . . 4 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → (∀z(z = y → φ) → ∀x∀z(z = y → φ)))
18	17	ex 423	. . 3 ⊢ (¬ ∀x x = z → (¬ ∀x x = y → (∀z(z = y → φ) → ∀x∀z(z = y → φ))))
19	5, 18	pm2.61i 156	. 2 ⊢ (¬ ∀x x = y → (∀z(z = y → φ) → ∀x∀z(z = y → φ)))
20		dvelimh.2	. . 3 ⊢ (ψ → ∀zψ)
21		dvelimh.3	. . 3 ⊢ (z = y → (φ ↔ ψ))
22	20, 21	equsalh 1961	. 2 ⊢ (∀z(z = y → φ) ↔ ψ)
23	22	albii 1566	. 2 ⊢ (∀x∀z(z = y → φ) ↔ ∀xψ)
24	19, 22, 23	3imtr3g 260	1 ⊢ (¬ ∀x x = y → (ψ → ∀xψ))

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

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:  dvelim  2016  dveeq1-o16  2188  dveel2ALT  2191