Theorem dvelimf-o 36870

Description: Proof of dvelimh 2450 that uses ax-c11 36828 but not ax-c15 36830, ax-c11n 36829, or ax-12 2173. Version of dvelimh 2450 using ax-c11 36828 instead of axc11 2430. (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 36838	. . . . 5 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) → ∀𝑧∀𝑧(𝑧 = 𝑦 → 𝜑))
2		ax-c11 36828	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧∀𝑧(𝑧 = 𝑦 → 𝜑) → ∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑)))
3	2	aecoms-o 36843	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑧∀𝑧(𝑧 = 𝑦 → 𝜑) → ∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑)))
4	1, 3	syl5 34	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑧(𝑧 = 𝑦 → 𝜑) → ∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑)))
5	4	a1d 25	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑦 → 𝜑) → ∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑))))
6		hbnae-o 36869	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑧)
7		hbnae-o 36869	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
8	6, 7	hban 2300	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑧(¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦))
9		hbnae-o 36869	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑧)
10		hbnae-o 36869	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
11	9, 10	hban 2300	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥(¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦))
12		ax-c9 36831	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
13	12	imp 406	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
14		dvelimf-o.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑥𝜑)
15	14	a1i 11	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝜑 → ∀𝑥𝜑))
16	11, 13, 15	hbimd 2298	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ((𝑧 = 𝑦 → 𝜑) → ∀𝑥(𝑧 = 𝑦 → 𝜑)))
17	8, 16	hbald 2170	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑧(𝑧 = 𝑦 → 𝜑) → ∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑)))
18	17	ex 412	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑦 → 𝜑) → ∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑))))
19	5, 18	pm2.61i 182	. 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 2420	. 2 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ 𝜓)
23	22	albii 1823	. 2 ⊢ (∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ ∀𝑥𝜓)
24	19, 22, 23	3imtr3g 294	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372  ax-c5 36824  ax-c4 36825  ax-c7 36826  ax-c10 36827  ax-c11 36828  ax-c9 36831

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788

This theorem is referenced by:  dveeq2-o  36874  dveeq1-o  36876  ax12el  36883