Theorem dvelimdf 2009

Description: Deduction form of dvelimf 2008. This version may be useful if we want to avoid ax-17 1519 and use ax-16 1807 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 1515	. . 3 ⊢ (𝜑 → ∀𝑧Ⅎ𝑥𝜓)
4		nfsb4t 2007	. . 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 1781	. . 3 ⊢ (𝜑 → ([𝑦 / 𝑧]𝜓 ↔ 𝜒))
10	6, 9	nfbidf 1532	. 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 1346  Ⅎwnf 1453  [wsb 1755

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756

This theorem is referenced by:  dvelimdc  2333