Theorem sbceqal

Description: Class version of one implication of equvelv . (Contributed by Andrew Salmon, 28-Jun-2011) (Proof shortened by SN, 26-Oct-2024)

	Ref	Expression
Assertion	sbceqal	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) → 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 = 𝐴 ↔ 𝐴 = 𝐴 ) )
2		eqeq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 = 𝐵 ↔ 𝐴 = 𝐵 ) )
3	1 2	imbi12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) ↔ ( 𝐴 = 𝐴 → 𝐴 = 𝐵 ) ) )
4		eqid	⊢ 𝐴 = 𝐴
5	4	a1bi	⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 = 𝐴 → 𝐴 = 𝐵 ) )
6	3 5	bitr4di	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) ↔ 𝐴 = 𝐵 ) )
7	6	spcgv	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) → 𝐴 = 𝐵 ) )