Theorem nfequid-o 39574

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-4 1836, ax-7 2035, ax-c9 39554, and ax-gen 1822. This shows that this can be proved without ax6 2422, even though Theorem equid 2039 cannot. A shorter proof using ax6 2422 is obtainable from equid 2039 and hbth 1830.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1995, which is used for the derivation of axc9 2420, unless we consider ax-c9 39554 the starting axiom rather than ax-13 2410. (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 39573	. 2 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
2	1	nf5i 2187	1 ⊢ Ⅎ𝑦 𝑥 = 𝑥

Syntax hints:  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-c9 39554

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811

This theorem is referenced by: (None)