Theorem hbequid 1524

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 1515 and ax-i12 1518 on top of (the FOL analogue of) modal logic KT. This shows that this can be proved without ax-i9 1541, even though Theorem equid 1712 cannot. A shorter proof using ax-i9 1541 is obtainable from equid 1712 and hbth 1474. (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

Step	Hyp	Ref	Expression
1		ax12or 1519	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 ∨ (∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)))
2		ax-8 1515	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝑥 = 𝑥))
3	2	pm2.43i 49	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑥)
4	3	alimi 1466	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
5	4	a1d 22	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
6		ax-4 1521	. . . 4 ⊢ (∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
7	5, 6	jaoi 717	. . 3 ⊢ ((∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
8	5, 7	jaoi 717	. 2 ⊢ ((∀𝑦 𝑦 = 𝑥 ∨ (∀𝑦 𝑦 = 𝑥 ∨ ∀𝑦(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))) → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
9	1, 8	ax-mp 5	1 ⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)

Syntax hints:   → wi 4   ∨ wo 709  ∀wal 1362   = wceq 1364

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 710  ax-5 1458  ax-gen 1460  ax-8 1515  ax-i12 1518  ax-4 1521

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  equveli  1770