Theorem nfequid-o 2161

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 uses only ax-5 1557, ax-8 1675, ax-12o 2142, and ax-gen 1546. This shows that this can be proved without ax9 1949, even though Theorem equid 1676 cannot. A shorter proof using ax9 1949 is obtainable from equid 1676 and hbth 1552.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1655, which is used for the derivation of ax12o 1934, unless we consider ax-12o 2142 the starting axiom rather than ax-12 1925. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nfequid-o	⊢ Ⅎy x = x

Proof of Theorem nfequid-o

Step	Hyp	Ref	Expression
1		hbequid 2160	. 2 ⊢ (x = x → ∀y x = x)
2	1	nfi 1551	1 ⊢ Ⅎy x = x

Syntax hints:  Ⅎwnf 1544

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 depends on definitions:  df-bi 177  df-nf 1545

This theorem is referenced by: (None)