Theorem dvelimdf 2450

Description: Deduction form of dvelimf 2449. Usage of this theorem is discouraged because it depends on ax-13 2373. (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 2201	. . 3 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
4		dvelimdf.2	. . . 4 ⊢ Ⅎ𝑧𝜑
5		dvelimdf.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑧𝜒)
6	4, 5	nfim1 2201	. . 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 276	. . 3 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
10	3, 6, 9	dvelimf 2449	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝜑 → 𝜒))
11		pm5.5 365	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜒) ↔ 𝜒))
12	1, 11	nfbidf 2226	. 2 ⊢ (𝜑 → (Ⅎ𝑥(𝜑 → 𝜒) ↔ Ⅎ𝑥𝜒))
13	10, 12	syl5ib 247	1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1540  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-11 2162  ax-12 2179  ax-13 2373

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791

This theorem is referenced by:  nfsb4t  2504  dvelimdc  2927