Theorem parteq12

Description: Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024)

	Ref	Expression
Assertion	parteq12	⊢ ( ( 𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ) → ( 𝑅 Part 𝐴 ↔ 𝑆 Part 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		parteq1	⊢ ( 𝑅 = 𝑆 → ( 𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴 ) )
2		parteq2	⊢ ( 𝐴 = 𝐵 → ( 𝑆 Part 𝐴 ↔ 𝑆 Part 𝐵 ) )
3	1 2	sylan9bb	⊢ ( ( 𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ) → ( 𝑅 Part 𝐴 ↔ 𝑆 Part 𝐵 ) )