Theorem equidqe 2173

Description: equid 1676 with existential quantifier without using ax-4 2135 or ax-17 1616. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.)

Assertion

Ref	Expression
equidqe	⊢ ¬ ∀y ¬ x = x

Proof of Theorem equidqe

Step	Hyp	Ref	Expression
1		ax9from9o 2148	. 2 ⊢ ¬ ∀y ¬ y = x
2		ax-8 1675	. . . . 5 ⊢ (y = x → (y = x → x = x))
3	2	pm2.43i 43	. . . 4 ⊢ (y = x → x = x)
4	3	con3i 127	. . 3 ⊢ (¬ x = x → ¬ y = x)
5	4	alimi 1559	. 2 ⊢ (∀y ¬ x = x → ∀y ¬ y = x)
6	1, 5	mto 167	1 ⊢ ¬ ∀y ¬ x = x

Syntax hints:  ¬ wn 3  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-8 1675  ax-6o 2137  ax-9o 2138

This theorem is referenced by:  ax4sp1  2174  equidq  2175