Theorem bj-hbaeb2

Description: Biconditional version of a form of hbae with commuted quantifiers, not requiring ax-11 . (Contributed by BJ, 12-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-hbaeb2	⊢ ( ∀ 𝑥 𝑥 = 𝑦 ↔ ∀ 𝑥 ∀ 𝑧 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		sp	⊢ ( ∀ 𝑧 𝑥 = 𝑦 → 𝑥 = 𝑦 )
12	11	alimi	⊢ ( ∀ 𝑥 ∀ 𝑧 𝑥 = 𝑦 → ∀ 𝑥 𝑥 = 𝑦 )
13	10 12	impbii	⊢ ( ∀ 𝑥 𝑥 = 𝑦 ↔ ∀ 𝑥 ∀ 𝑧 𝑥 = 𝑦 )