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Theorem equidq 34528
Description: equid 1985 with universal quantifier without using ax-c5 34487 or ax-5 1879. (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 34526 . 2 ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2 ax10fromc7 34499 . . 3 (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ ∀𝑦 𝑥 = 𝑥)
3 hbequid 34513 . . . 4 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
43con3i 150 . . 3 (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
52, 4alrimih 1791 . 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 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-c5 34487  ax-c4 34488  ax-c7 34489  ax-c10 34490  ax-c9 34494
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by: (None)
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