Theorem equidq

Description: equid with universal quantifier without using ax-c5 or ax-5 . (Contributed by NM, 13-Jan-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	equidq	⊢ ∀ 𝑦 𝑥 = 𝑥

Proof

Step	Hyp	Ref	Expression
1		equidqe	⊢ ¬ ∀ 𝑦 ¬ 𝑥 = 𝑥
2		ax10fromc7	⊢ ( ¬ ∀ 𝑦 𝑥 = 𝑥 → ∀ 𝑦 ¬ ∀ 𝑦 𝑥 = 𝑥 )
3		hbequid	⊢ ( 𝑥 = 𝑥 → ∀ 𝑦 𝑥 = 𝑥 )
4	3	con3i	⊢ ( ¬ ∀ 𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥 )
5	2 4	alrimih	⊢ ( ¬ ∀ 𝑦 𝑥 = 𝑥 → ∀ 𝑦 ¬ 𝑥 = 𝑥 )
6	1 5	mt3	⊢ ∀ 𝑦 𝑥 = 𝑥