Theorem bj-ssblem1

Description: A lemma for the definiens of df-sb . An instance of sp proved without it. Note: it has a common subproof with sbjust . (Contributed by BJ, 22-Dec-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ssblem1	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 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	spw	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )