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Theorem equidqe 1512
 Description: equid 1677 with some quantification and negation without using ax-4 1487 or ax-17 1506. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
Assertion
Ref Expression
equidqe ¬ ∀𝑦 ¬ 𝑥 = 𝑥

Proof of Theorem equidqe
StepHypRef Expression
1 ax-9 1511 . 2 ¬ ∀𝑦 ¬ 𝑦 = 𝑥
2 ax-8 1482 . . . . 5 (𝑦 = 𝑥 → (𝑦 = 𝑥𝑥 = 𝑥))
32pm2.43i 49 . . . 4 (𝑦 = 𝑥𝑥 = 𝑥)
43con3i 621 . . 3 𝑥 = 𝑥 → ¬ 𝑦 = 𝑥)
54alimi 1431 . 2 (∀𝑦 ¬ 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑦 = 𝑥)
61, 5mto 651 1 ¬ ∀𝑦 ¬ 𝑥 = 𝑥
 Colors of variables: wff set class Syntax hints:  ¬ wn 3  ∀wal 1329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie2 1470  ax-8 1482  ax-i9 1510 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337 This theorem is referenced by:  ax4sp1  1513
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