Theorem dvelimh 2450

Description: Version of dvelim 2451 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2372. Check out dvelimhw 2345 for a version requiring fewer axioms. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dvelimh.1	⊢ (𝜑 → ∀𝑥𝜑)
dvelimh.2	⊢ (𝜓 → ∀𝑧𝜓)
dvelimh.3	⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
dvelimh	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Proof of Theorem dvelimh

Step	Hyp	Ref	Expression
1		dvelimh.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	nf5i 2144	. . 3 ⊢ Ⅎ𝑥𝜑
3		dvelimh.2	. . . 4 ⊢ (𝜓 → ∀𝑧𝜓)
4	3	nf5i 2144	. . 3 ⊢ Ⅎ𝑧𝜓
5		dvelimh.3	. . 3 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
6	2, 4, 5	dvelimf 2448	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
7	6	nf5rd 2192	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

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

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:  dvelim  2451  dveeq1-o16  36877  dveel2ALT  36880  ax6e2nd  42067  ax6e2ndVD  42417  ax6e2ndALT  42439