Theorem nfequid-o 40104

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 1841, ax-7 2040, ax-c9 40084, and ax-gen 1827. This shows that this can be proved without ax6 2413, even though Theorem equid 2044 cannot. A shorter proof using ax6 2413 is obtainable from equid 2044 and hbth 1835.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 2000, which is used for the derivation of axc9 2411, unless we consider ax-c9 40084 the starting axiom rather than ax-13 2401. (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 40103	. 2 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
2	1	nf5i 2182	1 ⊢ Ⅎ𝑦 𝑥 = 𝑥

This proof depends on syntax axioms:  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-c9 40084

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816

This theorem is used by: (None)