Theorem uneq12d

Description: Equality deduction for the union of two classes. (Contributed by NM, 29-Sep-2004) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Hypotheses	uneq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	uneq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	uneq12d	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		uneq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		uneq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		uneq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 ) )