Theorem nfequid-o 39541

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 1832, ax-7 2031, ax-c9 39521, and ax-gen 1818. This shows that this can be proved without ax6 2418, even though Theorem equid 2035 cannot. A shorter proof using ax6 2418 is obtainable from equid 2035 and hbth 1826.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1991, which is used for the derivation of axc9 2416, unless we consider ax-c9 39521 the starting axiom rather than ax-13 2406. (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 39540	. 2 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
2	1	nf5i 2183	1 ⊢ Ⅎ𝑦 𝑥 = 𝑥

Syntax hints:  Ⅎwnf 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-c9 39521

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-nf 1807

This theorem is referenced by: (None)