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

Step	Hyp	Ref	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)
4	3	con3i 157	. . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
5	2, 4	alrimih 1830	. 2 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑥 = 𝑥)
6	1, 5	mt3 204	1 ⊢ ∀𝑦 𝑥 = 𝑥

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)