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Theorem dvelimv 2476
Description: Similar to dvelim 2475 with first hypothesis replaced by a distinct variable condition. Usage of this theorem is discouraged because it depends on ax-13 2392. Check out dvelimhw 2368 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 → (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 1912 . 2 (𝜑 → ∀𝑥𝜑)
2 dvelimv.1 . 2 (𝑧 = 𝑦 → (𝜑𝜓))
31, 2dvelim 2475 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786
This theorem is referenced by:  dveeq2ALT  2478  dveel1  2486  dveel2  2487  rgen2a  3223
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