Theorem dvelimv 2326

Description: Similar to dvelim 2325 with first hypothesis replaced by a distinct variable condition. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)

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 1827	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		dvelimv.1	. 2 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	dvelim 2325	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by:  dveeq2ALT  2328  dveel1  2358  dveel2  2359  rgen2a  2960