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Theorem hbequid 1513
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-8 1504 and ax-i12 1507 on top of (the FOL analogue of) modal logic KT. This shows that this can be proved without ax-i9 1530, even though Theorem equid 1701 cannot. A shorter proof using ax-i9 1530 is obtainable from equid 1701 and hbth 1463. (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 1508 . 2 (∀𝑦 𝑦 = 𝑥 ∨ (∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)))
2 ax-8 1504 . . . . . 6 (𝑦 = 𝑥 → (𝑦 = 𝑥𝑥 = 𝑥))
32pm2.43i 49 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑥)
43alimi 1455 . . . 4 (∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
54a1d 22 . . 3 (∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
6 ax-4 1510 . . . 4 (∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
75, 6jaoi 716 . . 3 ((∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
85, 7jaoi 716 . 2 ((∀𝑦 𝑦 = 𝑥 ∨ (∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
91, 8ax-mp 5 1 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 708  wal 1351   = wceq 1353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-gen 1449  ax-8 1504  ax-i12 1507  ax-4 1510
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  equveli  1759
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