Theorem dvelimALT 2133

Description: Version of dvelim 2016 that doesn't use ax-10 2140. (See dvelimh 1964 for a version that doesn't use ax-11 1746.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable groups:   ψ,z   x,z   y,z

Allowed substitution hints:   φ(x,y,z)   ψ(x,y)

Proof of Theorem dvelimALT

Step	Hyp	Ref	Expression
1		ax-17 1616	. . 3 ⊢ (¬ ∀x x = y → ∀z ¬ ∀x x = y)
2		ax16ALT 2047	. . . . 5 ⊢ (∀x x = z → ((z = y → φ) → ∀x(z = y → φ)))
3	2	a1d 22	. . . 4 ⊢ (∀x x = z → (¬ ∀x x = y → ((z = y → φ) → ∀x(z = y → φ))))
4		hbn1 1730	. . . . . . 7 ⊢ (¬ ∀x x = z → ∀x ¬ ∀x x = z)
5		hbn1 1730	. . . . . . 7 ⊢ (¬ ∀x x = y → ∀x ¬ ∀x x = y)
6	4, 5	hban 1828	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → ∀x(¬ ∀x x = z ∧ ¬ ∀x x = y))
7		ax12o 1934	. . . . . . 7 ⊢ (¬ ∀x x = z → (¬ ∀x x = y → (z = y → ∀x z = y)))
8	7	imp 418	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → (z = y → ∀x z = y))
9		dvelimALT.1	. . . . . . 7 ⊢ (φ → ∀xφ)
10	9	a1i 10	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → (φ → ∀xφ))
11	6, 8, 10	hbimd 1815	. . . . 5 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → ((z = y → φ) → ∀x(z = y → φ)))
12	11	ex 423	. . . 4 ⊢ (¬ ∀x x = z → (¬ ∀x x = y → ((z = y → φ) → ∀x(z = y → φ))))
13	3, 12	pm2.61i 156	. . 3 ⊢ (¬ ∀x x = y → ((z = y → φ) → ∀x(z = y → φ)))
14	1, 13	hbald 1740	. 2 ⊢ (¬ ∀x x = y → (∀z(z = y → φ) → ∀x∀z(z = y → φ)))
15		ax-17 1616	. . 3 ⊢ (ψ → ∀zψ)
16		dvelimALT.2	. . 3 ⊢ (z = y → (φ ↔ ψ))
17	15, 16	equsalh 1961	. 2 ⊢ (∀z(z = y → φ) ↔ ψ)
18	17	albii 1566	. 2 ⊢ (∀x∀z(z = y → φ) ↔ ∀xψ)
19	14, 17, 18	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  df-sb 1649

This theorem is referenced by:  dveeq2-o16  2185