Theorem difeq12d

Description: Equality deduction for class difference. (Contributed by FL, 29-May-2014)

	Ref	Expression
Hypotheses	difeq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	difeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	difeq12d	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐶 ) = ( 𝐵 ∖ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		difeq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		difeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	difeq1d	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐶 ) = ( 𝐵 ∖ 𝐶 ) )
4	2	difeq2d	⊢ ( 𝜑 → ( 𝐵 ∖ 𝐶 ) = ( 𝐵 ∖ 𝐷 ) )
5	3 4	eqtrd	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐶 ) = ( 𝐵 ∖ 𝐷 ) )