Theorem hbequid 2160

Description: Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x is technically free according to the traditional definition of free variable. (The proof does not use ax-9o 2138.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hbequid	⊢ (x = x → ∀y x = x)

Proof of Theorem hbequid

Step	Hyp	Ref	Expression
1		ax-12o 2142	. 2 ⊢ (¬ ∀y y = x → (¬ ∀y y = x → (x = x → ∀y x = x)))
2		ax-8 1675	. . . . 5 ⊢ (y = x → (y = x → x = x))
3	2	pm2.43i 43	. . . 4 ⊢ (y = x → x = x)
4	3	alimi 1559	. . 3 ⊢ (∀y y = x → ∀y x = x)
5	4	a1d 22	. 2 ⊢ (∀y y = x → (x = x → ∀y x = x))
6	1, 5, 5	pm2.61ii 157	1 ⊢ (x = x → ∀y x = x)

Syntax hints:   → wi 4  ∀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-12o 2142

This theorem is referenced by:  nfequid-o  2161  equidq  2175