Theorem bj-sb

Description: A weak variant of sbid2 not requiring ax-13 nor ax-10 . On top of Tarski's FOL, one implication requires only ax12v , and the other requires only sp . (Contributed by BJ, 25-May-2021)

	Ref	Expression
Assertion	bj-sb	⊢ ( 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax12v	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
2	1	equcoms	⊢ ( 𝑦 = 𝑥 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
3	2	com12	⊢ ( 𝜑 → ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
4	3	alrimiv	⊢ ( 𝜑 → ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
5		sp	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) )
6	5	com12	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → 𝜑 ) )
7	6	equcoms	⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → 𝜑 ) )
8	7	a2i	⊢ ( ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ( 𝑦 = 𝑥 → 𝜑 ) )
9	8	alimi	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ∀ 𝑦 ( 𝑦 = 𝑥 → 𝜑 ) )
10		bj-eqs	⊢ ( 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → 𝜑 ) )
11	9 10	sylibr	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → 𝜑 )
12	4 11	impbii	⊢ ( 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )