Theorem equidqe 39724

Description: equid 2041 with existential quantifier without using ax-c5 39685 or ax-5 1939. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
equidqe	⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥

Proof of Theorem equidqe

Step	Hyp	Ref	Expression
1		ax6fromc10 39698	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑦 = 𝑥
2		ax7 2045	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥))
3	2	pm2.43i 53	. . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥)
4	3	con3i 155	. . 3 ⊢ (¬ 𝑥 = 𝑥 → ¬ 𝑦 = 𝑥)
5	4	alimi 1840	. 2 ⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑦 = 𝑥)
6	1, 5	mto 200	1 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-c7 39687  ax-c10 39688

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809

This theorem is used by:  axc5sp1  39725  equidq  39726