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Theorem bnj1405 31424
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 4714 . 2 (𝑋 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑋𝐵)
31, 2sylib 210 1 (𝜑 → ∃𝑦𝐴 𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  wrex 3090   ciun 4710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-iun 4712
This theorem is referenced by:  bnj1408  31621  bnj1450  31635  bnj1501  31652
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