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Metamath Proof Explorer

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 ) )