Theorem rdgeq12

Description: Equality theorem for the recursive definition generator. (Contributed by Scott Fenton, 28-Apr-2012)

	Ref	Expression
Assertion	rdgeq12	⊢ ( ( 𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ) → rec ( 𝐹 , 𝐴 ) = rec ( 𝐺 , 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		rdgeq2	⊢ ( 𝐴 = 𝐵 → rec ( 𝐹 , 𝐴 ) = rec ( 𝐹 , 𝐵 ) )
2		rdgeq1	⊢ ( 𝐹 = 𝐺 → rec ( 𝐹 , 𝐵 ) = rec ( 𝐺 , 𝐵 ) )
3	1 2	sylan9eqr	⊢ ( ( 𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ) → rec ( 𝐹 , 𝐴 ) = rec ( 𝐺 , 𝐵 ) )