Theorem equidq 2175

Description: equid 1676 with universal quantifier without using ax-4 2135 or ax-17 1616. (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 2173	. 2 ⊢ ¬ ∀y ¬ x = x
2		ax6 2147	. . 3 ⊢ (¬ ∀y x = x → ∀y ¬ ∀y x = x)
3		hbequid 2160	. . . 4 ⊢ (x = x → ∀y x = x)
4	3	con3i 127	. . 3 ⊢ (¬ ∀y x = x → ¬ x = x)
5	2, 4	alrimih 1565	. 2 ⊢ (¬ ∀y x = x → ∀y ¬ x = x)
6	1, 5	mt3 171	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-4 2135  ax-5o 2136  ax-6o 2137  ax-9o 2138  ax-12o 2142

This theorem is referenced by: (None)