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Theorem dveel2ALT 34747
Description: Alternate proof of dveel2 2518 using ax-c16 34700 instead of ax-5 1991. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveel2ALT (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveel2ALT
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax5el 34745 . 2 (𝑧𝑤 → ∀𝑥 𝑧𝑤)
2 ax5el 34745 . 2 (𝑧𝑦 → ∀𝑤 𝑧𝑦)
3 elequ2 2159 . 2 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
41, 2, 3dvelimh 2486 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-c14 34699  ax-c16 34700
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858
This theorem is referenced by: (None)
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