Theorem dvelimf-o 2180

Description: Proof of dvelimh 1964 that uses ax-10o 2139 but not ax-11o 2141, ax-10 2140, or ax-11 1746. Version of dvelimh 1964 using ax-10o 2139 instead of ax10o 1952. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
dvelimf-o	⊢ (¬ ∀x x = y → (ψ → ∀xψ))

Proof of Theorem dvelimf-o

Step	Hyp	Ref	Expression
1		hba1-o 2149	. . . . 5 ⊢ (∀z(z = y → φ) → ∀z∀z(z = y → φ))
2		ax-10o 2139	. . . . . 6 ⊢ (∀z z = x → (∀z∀z(z = y → φ) → ∀x∀z(z = y → φ)))
3	2	aecoms-o 2152	. . . . 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-o 2179	. . . . . 6 ⊢ (¬ ∀x x = z → ∀z ¬ ∀x x = z)
7		hbnae-o 2179	. . . . . 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-o 2179	. . . . . . 7 ⊢ (¬ ∀x x = z → ∀x ¬ ∀x x = z)
10		hbnae-o 2179	. . . . . . 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		ax-12o 2142	. . . . . . 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		dvelimf-o.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		dvelimf-o.2	. . 3 ⊢ (ψ → ∀zψ)
21		dvelimf-o.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  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142

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:  dveeq2-o  2184  dveeq1-o  2187  ax11el  2194