Theorem dvelimdf 2004

Description: Deduction form of dvelimf 2003. This version may be useful if we want to avoid ax-17 1514 and use ax-16 1802 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)

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.2	. . . 4 ⊢ Ⅎ𝑧𝜑
2		dvelimdf.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3	1, 2	alrimi 1510	. . 3 ⊢ (𝜑 → ∀𝑧Ⅎ𝑥𝜓)
4		nfsb4t 2002	. . 3 ⊢ (∀𝑧Ⅎ𝑥𝜓 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑧]𝜓))
5	3, 4	syl 14	. 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑧]𝜓))
6		dvelimdf.1	. . 3 ⊢ Ⅎ𝑥𝜑
7		dvelimdf.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑧𝜒)
8		dvelimdf.5	. . . 4 ⊢ (𝜑 → (𝑧 = 𝑦 → (𝜓 ↔ 𝜒)))
9	1, 7, 8	sbied 1776	. . 3 ⊢ (𝜑 → ([𝑦 / 𝑧]𝜓 ↔ 𝜒))
10	6, 9	nfbidf 1527	. 2 ⊢ (𝜑 → (Ⅎ𝑥[𝑦 / 𝑧]𝜓 ↔ Ⅎ𝑥𝜒))
11	5, 10	sylibd 148	1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  ∀wal 1341  Ⅎwnf 1448  [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751

This theorem is referenced by:  dvelimdc  2329