Theorem bnj1405 35033

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 4928	. 2 ⊢ (𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵)
3	1, 2	sylib 220	1 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2121  ∃wrex 3065  ∪ ciun 4924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rex 3066  df-v 3435  df-iun 4926

This theorem is referenced by:  bnj1408  35233  bnj1450  35247  bnj1501  35264