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Metamath Proof Explorer

Theorem uneq2

Description: Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993)

Ref Expression
Assertion uneq2 ( 𝐴 = 𝐵 → ( 𝐶 ∪ 𝐴 ) = ( 𝐶 ∪ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 uneq1 ⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) )
2 uncom ⊢ ( 𝐶 ∪ 𝐴 ) = ( 𝐴 ∪ 𝐶 )
3 uncom ⊢ ( 𝐶 ∪ 𝐵 ) = ( 𝐵 ∪ 𝐶 )
4 1 2 3 3eqtr4g ⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∪ 𝐴 ) = ( 𝐶 ∪ 𝐵 ) )