Theorem dvelimfALT2 1828

Description: Proof of dvelimf 2027 using dveeq2 1826 (shown as the last hypothesis) instead of ax12 1523. This shows that ax12 1523 could be replaced by dveeq2 1826 (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 1537	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2		hbn1 1663	. . . 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 1584	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑧 = 𝑦 → 𝜑) → ∀𝑥(𝑧 = 𝑦 → 𝜑)))
7	1, 6	hbald 1502	. 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 1737	. 2 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ 𝜓)
11	10	albii 1481	. 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 1362   = wceq 1364

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 615  ax-in2 616  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370

This theorem is referenced by: (None)