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Theorem hbequid 34718
 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 does not use ax-c10 34695.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbequid (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)

Proof of Theorem hbequid
StepHypRef Expression
1 ax-c9 34699 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)))
2 ax7 2101 . . . . 5 (𝑦 = 𝑥 → (𝑦 = 𝑥𝑥 = 𝑥))
32pm2.43i 52 . . . 4 (𝑦 = 𝑥𝑥 = 𝑥)
43alimi 1887 . . 3 (∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
54a1d 25 . 2 (∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
61, 5, 5pm2.61ii 177 1 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-c9 34699 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853 This theorem is referenced by:  nfequid-o  34719  equidq  34733
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