Theorem hbae

Description: All variables are effectively bound in an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker hbaev when possible. (Contributed by NM, 13-May-1993) (Proof shortened by Wolf Lammen, 21-Apr-2018) (New usage is discouraged.)

	Ref	Expression
Assertion	hbae	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 ∀ 𝑥 𝑥 = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		sp	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦 )
2		axc9	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) ) )
3	1 2	syl7	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) ) )
4		axc11r	⊢ ( ∀ 𝑧 𝑧 = 𝑥 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
5		axc11	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑦 𝑥 = 𝑦 ) )
6	5	pm2.43i	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑦 𝑥 = 𝑦 )
7		axc11r	⊢ ( ∀ 𝑧 𝑧 = 𝑦 → ( ∀ 𝑦 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
8	6 7	syl5	⊢ ( ∀ 𝑧 𝑧 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
9	3 4 8	pm2.61ii	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 )
10	9	axc4i	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑥 ∀ 𝑧 𝑥 = 𝑦 )
11		ax-11	⊢ ( ∀ 𝑥 ∀ 𝑧 𝑥 = 𝑦 → ∀ 𝑧 ∀ 𝑥 𝑥 = 𝑦 )
12	10 11	syl	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 ∀ 𝑥 𝑥 = 𝑦 )