Theorem dvelimv 2445

Description: Similar to dvelim 2444 with first hypothesis replaced by a distinct variable condition. Usage of this theorem is discouraged because it depends on ax-13 2365. Check out dvelimhw 2335 for a version requiring fewer axioms. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dvelimv.1	⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable groups:   𝜑,𝑥   𝜓,𝑧

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝜓(𝑥,𝑦)

Proof of Theorem dvelimv

Step	Hyp	Ref	Expression
1		ax-5 1905	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		dvelimv.1	. 2 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	dvelim 2444	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778

This theorem is referenced by:  dveeq2ALT  2447  dveel1  2454  dveel2  2455  rgen2a  3361