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Theorem nfequid-o 34717
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 1886, ax-7 2090, ax-c9 34697, and ax-gen 1871. This shows that this can be proved without ax6 2396, even though the theorem equid 2094 cannot be. A shorter proof using ax6 2396 is obtainable from equid 2094 and hbth 1878.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 2055, which is used for the derivation of axc9 2447, unless we consider ax-c9 34697 the starting axiom rather than ax-13 2391. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Ref Expression
nfequid-o 𝑦 𝑥 = 𝑥

Proof of Theorem nfequid-o
StepHypRef Expression
1 hbequid 34716 . 2 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
21nf5i 2173 1 𝑦 𝑥 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  wnf 1857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-c9 34697
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-nf 1859
This theorem is referenced by: (None)
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