Theorem sbjust

Description: Justification theorem for df-sb proved from Tarski's FOL axiom schemes. (Contributed by BJ, 22-Jan-2023)

	Ref	Expression
Assertion	sbjust	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ∀ 𝑧 ( 𝑧 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑧 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		equequ1	⊢ ( 𝑦 = 𝑧 → ( 𝑦 = 𝑡 ↔ 𝑧 = 𝑡 ) )
2		equequ2	⊢ ( 𝑦 = 𝑧 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝑧 ) )
3	2	imbi1d	⊢ ( 𝑦 = 𝑧 → ( ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( 𝑥 = 𝑧 → 𝜑 ) ) )
4	3	albidv	⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑧 → 𝜑 ) ) )
5	1 4	imbi12d	⊢ ( 𝑦 = 𝑧 → ( ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ( 𝑧 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑧 → 𝜑 ) ) ) )
6	5	cbvalvw	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ↔ ∀ 𝑧 ( 𝑧 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑧 → 𝜑 ) ) )