Theorem nfequid-o 34195

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 1737, ax-7 1935, ax-c9 34175, and ax-gen 1722. This shows that this can be proved without ax6 2251, even though the theorem equid 1939 cannot be. A shorter proof using ax6 2251 is obtainable from equid 1939 and hbth 1729.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1889, which is used for the derivation of axc9 2302, unless we consider ax-c9 34175 the starting axiom rather than ax-13 2246. (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 34194	. 2 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
2	1	nf5i 2024	1 ⊢ Ⅎ𝑦 𝑥 = 𝑥

Syntax hints:  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-c9 34175

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)