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Metamath Proof Explorer

Theorem bnj1316

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1316.1 ⊢ ( 𝑦 ∈ 𝐴 → ∀ 𝑥 𝑦 ∈ 𝐴 )
bnj1316.2 ⊢ ( 𝑦 ∈ 𝐵 → ∀ 𝑥 𝑦 ∈ 𝐵 )
Assertion bnj1316 ( 𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )

Proof

Step Hyp Ref Expression
1 bnj1316.1 ⊢ ( 𝑦 ∈ 𝐴 → ∀ 𝑥 𝑦 ∈ 𝐴 )
2 bnj1316.2 ⊢ ( 𝑦 ∈ 𝐵 → ∀ 𝑥 𝑦 ∈ 𝐵 )
3 1 nfcii ⊢ Ⅎ 𝑥 𝐴
4 2 nfcii ⊢ Ⅎ 𝑥 𝐵
5 3 4 nfeq ⊢ Ⅎ 𝑥 𝐴 = 𝐵
6 5 nf5ri ⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 𝐴 = 𝐵 )
7 6 bnj956 ⊢ ( 𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )