Theorem csbeq12

Description: Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018)

	Ref	Expression
Assertion	csbeq12	⊢ ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥 𝐶 = 𝐷 ) → ⦋ 𝐴 / 𝑥 ⦌ 𝐶 = ⦋ 𝐵 / 𝑥 ⦌ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		csbeq2	⊢ ( ∀ 𝑥 𝐶 = 𝐷 → ⦋ 𝐴 / 𝑥 ⦌ 𝐶 = ⦋ 𝐴 / 𝑥 ⦌ 𝐷 )
2		csbeq1	⊢ ( 𝐴 = 𝐵 → ⦋ 𝐴 / 𝑥 ⦌ 𝐷 = ⦋ 𝐵 / 𝑥 ⦌ 𝐷 )
3	1 2	sylan9eqr	⊢ ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥 𝐶 = 𝐷 ) → ⦋ 𝐴 / 𝑥 ⦌ 𝐶 = ⦋ 𝐵 / 𝑥 ⦌ 𝐷 )