Theorem bnj1405 34828

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 4999	. 2 ⊢ (𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵)
3	1, 2	sylib 218	1 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2105  ∃wrex 3067  ∪ ciun 4995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rex 3068  df-v 3479  df-iun 4997

This theorem is referenced by:  bnj1408  35028  bnj1450  35042  bnj1501  35059