Theorem bnj1405 33678

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

Hypothesis

Ref	Expression
bnj1405.1	⊢ (𝜑 → 𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵)

Assertion

Ref	Expression
bnj1405	⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵)

Distinct variable group:   𝑦,𝑋

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem bnj1405

Step	Hyp	Ref	Expression
1		bnj1405.1	. 2 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵)
2		eliun 4994	. 2 ⊢ (𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵)
3	1, 2	sylib 217	1 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2106  ∃wrex 3069  ∪ ciun 4990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-v 3475  df-iun 4992

This theorem is referenced by:  bnj1408  33878  bnj1450  33892  bnj1501  33909