Theorem dvelimfALT2 1817

Description: Proof of dvelimf 2015 using dveeq2 1815 (shown as the last hypothesis) instead of ax12 1512. This shows that ax12 1512 could be replaced by dveeq2 1815 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)

Hypotheses

Ref	Expression
dvelimfALT2.1	⊢ (𝜑 → ∀𝑥𝜑)
dvelimfALT2.2	⊢ (𝜓 → ∀𝑧𝜓)
dvelimfALT2.3	⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
dvelimfALT2.4	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Assertion

Ref	Expression
dvelimfALT2	⊢ (¬ ∀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 dvelimfALT2

Step	Hyp	Ref	Expression
1		ax-17 1526	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2		hbn1 1652	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
3		dvelimfALT2.4	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
4		dvelimfALT2.1	. . . . 5 ⊢ (𝜑 → ∀𝑥𝜑)
5	4	a1i 9	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
6	2, 3, 5	hbimd 1573	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑧 = 𝑦 → 𝜑) → ∀𝑥(𝑧 = 𝑦 → 𝜑)))
7	1, 6	hbald 1491	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑦 → 𝜑) → ∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑)))
8		dvelimfALT2.2	. . 3 ⊢ (𝜓 → ∀𝑧𝜓)
9		dvelimfALT2.3	. . 3 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
10	8, 9	equsalh 1726	. 2 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ 𝜓)
11	10	albii 1470	. 2 ⊢ (∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ ∀𝑥𝜓)
12	7, 10, 11	3imtr3g 204	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ∀wal 1351   = wceq 1353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359

This theorem is referenced by: (None)