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Theorem dvelimv 2369
Description: Similar to dvelim 2368 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 → (𝜑𝜓))
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
StepHypRef Expression
1 ax-5 1879 . 2 (𝜑 → ∀𝑥𝜑)
2 dvelimv.1 . 2 (𝑧 = 𝑦 → (𝜑𝜓))
31, 2dvelim 2368 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750
This theorem is referenced by:  dveeq2ALT  2371  dveel1  2398  dveel2  2399  rgen2a  3006
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