Theorem equidqe 1543

Description: equid 1712 with some quantification and negation without using ax-4 1521 or ax-17 1537. (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 1542	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑦 = 𝑥
2		ax-8 1515	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥))
3	2	pm2.43i 49	. . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥)
4	3	con3i 633	. . 3 ⊢ (¬ 𝑥 = 𝑥 → ¬ 𝑦 = 𝑥)
5	4	alimi 1466	. 2 ⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑦 = 𝑥)
6	1, 5	mto 663	1 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥

Syntax hints:  ¬ wn 3  ∀wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie2 1505  ax-8 1515  ax-i9 1541

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370

This theorem is referenced by:  ax4sp1  1544