Theorem dvelimfALT2 1746

Description: Proof of dvelimf 1940 using dveeq2 1744 (shown as the last hypothesis) instead of ax-12 1448. This shows that ax-12 1448 could be replaced by dveeq2 1744 (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 1465	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2		hbn1 1588	. . . 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 1511	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑧 = 𝑦 → 𝜑) → ∀𝑥(𝑧 = 𝑦 → 𝜑)))
7	1, 6	hbald 1426	. 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 1662	. 2 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ 𝜓)
11	10	albii 1405	. 2 ⊢ (∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ ∀𝑥𝜓)
12	7, 10, 11	3imtr3g 203	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  ∀wal 1288   = wceq 1290

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296

This theorem is referenced by: (None)