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Theorem equidq 36581
Description: equid 2024 with universal quantifier without using ax-c5 36540 or ax-5 1917. (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 36579 . 2 ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2 ax10fromc7 36552 . . 3 (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ ∀𝑦 𝑥 = 𝑥)
3 hbequid 36566 . . . 4 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
43con3i 157 . . 3 (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
52, 4alrimih 1830 . 2 (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑥 = 𝑥)
61, 5mt3 204 1 𝑦 𝑥 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-c5 36540  ax-c4 36541  ax-c7 36542  ax-c10 36543  ax-c9 36547
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787
This theorem is referenced by: (None)
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