Theorem equidq 38947

Description: equid 2012 with universal quantifier without using ax-c5 38906 or ax-5 1910. (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 38945	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2		ax10fromc7 38918	. . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ ∀𝑦 𝑥 = 𝑥)
3		hbequid 38932	. . . 4 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
4	3	con3i 154	. . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
5	2, 4	alrimih 1824	. 2 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑥 = 𝑥)
6	1, 5	mt3 201	1 ⊢ ∀𝑦 𝑥 = 𝑥

Syntax hints:  ¬ wn 3  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-c5 38906  ax-c4 38907  ax-c7 38908  ax-c10 38909  ax-c9 38913

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by: (None)