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Metamath Proof Explorer

Theorem uneq12i

Description: Equality inference for the union of two classes. (Contributed by NM, 12-Aug-2004) (Proof shortened by Eric Schmidt, 26-Jan-2007)

Ref Expression
Hypotheses uneq1i.1 ⊢ 𝐴 = 𝐵
uneq12i.2 ⊢ 𝐶 = 𝐷
Assertion uneq12i ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 )

Proof

Step Hyp Ref Expression
1 uneq1i.1 ⊢ 𝐴 = 𝐵
2 uneq12i.2 ⊢ 𝐶 = 𝐷
3 uneq12 ⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 ) )
4 1 2 3 mp2an ⊢ ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 )