Theorem equidqe 1513

Description: equid 1678 with some quantification and negation without using ax-4 1488 or ax-17 1507. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)

Assertion

Ref	Expression
equidqe	⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥

Proof of Theorem equidqe

Step	Hyp	Ref	Expression
1		ax-9 1512	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑦 = 𝑥
2		ax-8 1483	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥))
3	2	pm2.43i 49	. . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥)
4	3	con3i 622	. . 3 ⊢ (¬ 𝑥 = 𝑥 → ¬ 𝑦 = 𝑥)
5	4	alimi 1432	. 2 ⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑦 = 𝑥)
6	1, 5	mto 652	1 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥

Syntax hints:  ¬ wn 3  ∀wal 1330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie2 1471  ax-8 1483  ax-i9 1511

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338

This theorem is referenced by:  ax4sp1  1514