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Metamath Proof Explorer

Theorem iuneq1d

Description: Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015)

Ref Expression
Hypothesis iuneq1d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion iuneq1d ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )

Proof

Step Hyp Ref Expression
1 iuneq1d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 iuneq1 ⊢ ( 𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )
3 1 2 syl ⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )