Theorem dvelimdf 2481

Description: Deduction form of dvelimf 2480. Usage of this theorem is discouraged because it depends on ax-13 2404. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dvelimdf.1	⊢ Ⅎ𝑥𝜑
dvelimdf.2	⊢ Ⅎ𝑧𝜑
dvelimdf.3	⊢ (𝜑 → Ⅎ𝑥𝜓)
dvelimdf.4	⊢ (𝜑 → Ⅎ𝑧𝜒)
dvelimdf.5	⊢ (𝜑 → (𝑧 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
dvelimdf	⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))

Proof of Theorem dvelimdf

Step	Hyp	Ref	Expression
1		dvelimdf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		dvelimdf.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3	1, 2	nfim1 2235	. . 3 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
4		dvelimdf.2	. . . 4 ⊢ Ⅎ𝑧𝜑
5		dvelimdf.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑧𝜒)
6	4, 5	nfim1 2235	. . 3 ⊢ Ⅎ𝑧(𝜑 → 𝜒)
7		dvelimdf.5	. . . . 5 ⊢ (𝜑 → (𝑧 = 𝑦 → (𝜓 ↔ 𝜒)))
8	7	com12 33	. . . 4 ⊢ (𝑧 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒)))
9	8	pm5.74d 276	. . 3 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
10	3, 6, 9	dvelimf 2480	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝜑 → 𝜒))
11		pm5.5 364	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜒) ↔ 𝜒))
12	1, 11	nfbidf 2260	. 2 ⊢ (𝜑 → (Ⅎ𝑥(𝜑 → 𝜒) ↔ Ⅎ𝑥𝜒))
13	10, 12	imbitrid 247	1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814

This theorem is referenced by:  nfsb4t  2531  dvelimdc  2949