Theorem dvelimh 2482

Description: Version of dvelim 2483 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2404. Check out dvelimhw 2378 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 2186	. . 3 ⊢ Ⅎ𝑥𝜑
3		dvelimh.2	. . . 4 ⊢ (𝜓 → ∀𝑧𝜓)
4	3	nf5i 2186	. . 3 ⊢ Ⅎ𝑧𝜓
5		dvelimh.3	. . 3 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
6	2, 4, 5	dvelimf 2480	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
7	6	nf5rd 2236	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817

This theorem is used by:  dvelim  2483  dveeq1-o16  40259  dveel2ALT  40262  ax6e2nd  45801  ax6e2ndVD  46150  ax6e2ndALT  46172