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Theorem equidq 34732
 Description: equid 2097 with universal quantifier without using ax-c5 34691 or ax-5 1991. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equidq 𝑦 𝑥 = 𝑥

Proof of Theorem equidq
StepHypRef Expression
1 equidqe 34730 . 2 ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2 ax10fromc7 34703 . . 3 (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ ∀𝑦 𝑥 = 𝑥)
3 hbequid 34717 . . . 4 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
43con3i 151 . . 3 (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
52, 4alrimih 1899 . 2 (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑥 = 𝑥)
61, 5mt3 192 1 𝑦 𝑥 = 𝑥
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ∀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-c5 34691  ax-c4 34692  ax-c7 34693  ax-c10 34694  ax-c9 34698 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853 This theorem is referenced by: (None)
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