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Theorem bnj1405 34850
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1405.1 (𝜑𝑋 𝑦𝐴 𝐵)
Assertion
Ref Expression
bnj1405 (𝜑 → ∃𝑦𝐴 𝑋𝐵)
Distinct variable group:   𝑦,𝑋
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem bnj1405
StepHypRef Expression
1 bnj1405.1 . 2 (𝜑𝑋 𝑦𝐴 𝐵)
2 eliun 4995 . 2 (𝑋 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑋𝐵)
31, 2sylib 218 1 (𝜑 → ∃𝑦𝐴 𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wrex 3070   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482  df-iun 4993
This theorem is referenced by:  bnj1408  35050  bnj1450  35064  bnj1501  35081
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