Theorem dvelimv 2474

Description: Similar to dvelim 2473 with first hypothesis replaced by a distinct variable condition. Usage of this theorem is discouraged because it depends on ax-13 2390. Check out dvelimhw 2366 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 1911	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		dvelimv.1	. 2 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	dvelim 2473	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785

This theorem is referenced by:  dveeq2ALT  2476  dveel1  2484  dveel2  2485  rgen2a  3231