Theorem equidq 39739

Description: equid 2045 with universal quantifier without using ax-c5 39698 or ax-5 1943. (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 39737	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2		ax10fromc7 39710	. . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ ∀𝑦 𝑥 = 𝑥)
3		hbequid 39724	. . . 4 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
4	3	con3i 155	. . 3 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
5	2, 4	alrimih 1857	. 2 ⊢ (¬ ∀𝑦 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑥 = 𝑥)
6	1, 5	mt3 204	1 ⊢ ∀𝑦 𝑥 = 𝑥

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-c5 39698  ax-c4 39699  ax-c7 39700  ax-c10 39701  ax-c9 39705

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)