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Theorem hbequid 1449
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-5 1379, ax-8 1438, ax-12 1445, and ax-gen 1381. This shows that this can be proved without ax-9 1467, even though the theorem equid 1632 cannot be. A shorter proof using ax-9 1467 is obtainable from equid 1632 and hbth 1395.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.)
Assertion
Ref Expression
hbequid (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)

Proof of Theorem hbequid
StepHypRef Expression
1 ax12or 1446 . 2 (∀𝑦 𝑦 = 𝑥 ∨ (∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)))
2 ax-8 1438 . . . . . 6 (𝑦 = 𝑥 → (𝑦 = 𝑥𝑥 = 𝑥))
32pm2.43i 48 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑥)
43alimi 1387 . . . 4 (∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
54a1d 22 . . 3 (∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
6 ax-4 1443 . . . 4 (∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
75, 6jaoi 669 . . 3 ((∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
85, 7jaoi 669 . 2 ((∀𝑦 𝑦 = 𝑥 ∨ (∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
91, 8ax-mp 7 1 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662  wal 1285   = wceq 1287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-gen 1381  ax-8 1438  ax-i12 1441  ax-4 1443
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  equveli  1686
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