Theorem nfequid-o 38933

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 1809, ax-7 2008, ax-c9 38913, and ax-gen 1795. This shows that this can be proved without ax6 2389, even though Theorem equid 2012 cannot. A shorter proof using ax6 2389 is obtainable from equid 2012 and hbth 1803.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1968, which is used for the derivation of axc9 2387, unless we consider ax-c9 38913 the starting axiom rather than ax-13 2377. (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 38932	. 2 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
2	1	nf5i 2147	1 ⊢ Ⅎ𝑦 𝑥 = 𝑥

Syntax hints:  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-c9 38913

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)