Theorem dvelimdf 2449

Description: Deduction form of dvelimf 2448. Usage of this theorem is discouraged because it depends on ax-13 2372. (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 2195	. . 3 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
4		dvelimdf.2	. . . 4 ⊢ Ⅎ𝑧𝜑
5		dvelimdf.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑧𝜒)
6	4, 5	nfim1 2195	. . 3 ⊢ Ⅎ𝑧(𝜑 → 𝜒)
7		dvelimdf.5	. . . . 5 ⊢ (𝜑 → (𝑧 = 𝑦 → (𝜓 ↔ 𝜒)))
8	7	com12 32	. . . 4 ⊢ (𝑧 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒)))
9	8	pm5.74d 272	. . 3 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
10	3, 6, 9	dvelimf 2448	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝜑 → 𝜒))
11		pm5.5 361	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜒) ↔ 𝜒))
12	1, 11	nfbidf 2220	. 2 ⊢ (𝜑 → (Ⅎ𝑥(𝜑 → 𝜒) ↔ Ⅎ𝑥𝜒))
13	10, 12	syl5ib 243	1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1537  Ⅎwnf 1787

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

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:  nfsb4t  2503  dvelimdc  2933