This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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