Theorem nfequid-o 34014

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 1735, ax-7 1933, ax-c9 33994, and ax-gen 1720. This shows that this can be proved without ax6 2249, even though the theorem equid 1937 cannot be. A shorter proof using ax6 2249 is obtainable from equid 1937 and hbth 1727.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1887, which is used for the derivation of axc9 2300, unless we consider ax-c9 33994 the starting axiom rather than ax-13 2244. (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 34013	. 2 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
2	1	nf5i 2022	1 ⊢ Ⅎ𝑦 𝑥 = 𝑥

Syntax hints:  Ⅎwnf 1706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-c9 33994

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-nf 1708

This theorem is referenced by: (None)