Theorem sbeqalb

Description: Theorem *14.121 in WhiteheadRussell p. 185. (Contributed by Andrew Salmon, 28-Jun-2011) (Proof shortened by Wolf Lammen, 9-May-2013)

	Ref	Expression
Assertion	sbeqalb	⊢ ( 𝐴 ∈ 𝑉 → ( ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐴 ) ∧ ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐵 ) ) → 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		bibi1	⊢ ( ( 𝜑 ↔ 𝑥 = 𝐴 ) → ( ( 𝜑 ↔ 𝑥 = 𝐵 ) ↔ ( 𝑥 = 𝐴 ↔ 𝑥 = 𝐵 ) ) )
2	1	biimpa	⊢ ( ( ( 𝜑 ↔ 𝑥 = 𝐴 ) ∧ ( 𝜑 ↔ 𝑥 = 𝐵 ) ) → ( 𝑥 = 𝐴 ↔ 𝑥 = 𝐵 ) )
3	2	biimpd	⊢ ( ( ( 𝜑 ↔ 𝑥 = 𝐴 ) ∧ ( 𝜑 ↔ 𝑥 = 𝐵 ) ) → ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) )
4	3	alanimi	⊢ ( ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐴 ) ∧ ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐵 ) ) → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) )
5		sbceqal	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 = 𝐵 ) → 𝐴 = 𝐵 ) )
6	4 5	syl5	⊢ ( 𝐴 ∈ 𝑉 → ( ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐴 ) ∧ ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐵 ) ) → 𝐴 = 𝐵 ) )