Theorem dvelimhw 1849

Description: Proof of dvelimh 1964 without using ax-12 1925 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.)

Hypotheses

Ref	Expression
dvelimhw.1	⊢ (φ → ∀xφ)
dvelimhw.2	⊢ (ψ → ∀zψ)
dvelimhw.3	⊢ (z = y → (φ ↔ ψ))
dvelimhw.4	⊢ (¬ ∀x x = y → (y = z → ∀x y = z))

Assertion

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

Distinct variable groups:   x,z   y,z

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

Proof of Theorem dvelimhw

Step	Hyp	Ref	Expression
1		ax-17 1616	. . 3 ⊢ (¬ ∀x x = y → ∀z ¬ ∀x x = y)
2		hbn1 1730	. . . 4 ⊢ (¬ ∀x x = y → ∀x ¬ ∀x x = y)
3		equcomi 1679	. . . . 5 ⊢ (z = y → y = z)
4		dvelimhw.4	. . . . 5 ⊢ (¬ ∀x x = y → (y = z → ∀x y = z))
5		equcomi 1679	. . . . . 6 ⊢ (y = z → z = y)
6	5	alimi 1559	. . . . 5 ⊢ (∀x y = z → ∀x z = y)
7	3, 4, 6	syl56 30	. . . 4 ⊢ (¬ ∀x x = y → (z = y → ∀x z = y))
8		dvelimhw.1	. . . . 5 ⊢ (φ → ∀xφ)
9	8	a1i 10	. . . 4 ⊢ (¬ ∀x x = y → (φ → ∀xφ))
10	2, 7, 9	hbimd 1815	. . 3 ⊢ (¬ ∀x x = y → ((z = y → φ) → ∀x(z = y → φ)))
11	1, 10	hbald 1740	. 2 ⊢ (¬ ∀x x = y → (∀z(z = y → φ) → ∀x∀z(z = y → φ)))
12		dvelimhw.2	. . 3 ⊢ (ψ → ∀zψ)
13		dvelimhw.3	. . 3 ⊢ (z = y → (φ ↔ ψ))
14	12, 13	equsalhw 1838	. 2 ⊢ (∀z(z = y → φ) ↔ ψ)
15	14	albii 1566	. 2 ⊢ (∀x∀z(z = y → φ) ↔ ∀xψ)
16	11, 14, 15	3imtr3g 260	1 ⊢ (¬ ∀x x = y → (ψ → ∀xψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀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

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax12olem6  1932