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Theorem equidq 38396
Description: equid 2008 with universal quantifier without using ax-c5 38355 or ax-5 1906. (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 38394 . 2 ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2 ax10fromc7 38367 . . 3 (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ ∀𝑦 𝑥 = 𝑥)
3 hbequid 38381 . . . 4 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
43con3i 154 . . 3 (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
52, 4alrimih 1819 . 2 (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑥 = 𝑥)
61, 5mt3 200 1 𝑦 𝑥 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-c5 38355  ax-c4 38356  ax-c7 38357  ax-c10 38358  ax-c9 38362
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775
This theorem is referenced by: (None)
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