Theorem bnj1322

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Assertion	bnj1322	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eqimss	⊢ ( 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵 )
2		dfss2	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 𝐴 )
3	1 2	sylib	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ 𝐵 ) = 𝐴 )