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Theorem bnj1405 29477
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1405.1  |-  ( ph  ->  X  e.  U_ y  e.  A  B )
Assertion
Ref Expression
bnj1405  |-  ( ph  ->  E. y  e.  A  X  e.  B )
Distinct variable group:    y, X
Allowed substitution hints:    ph( y)    A( y)    B( y)

Proof of Theorem bnj1405
StepHypRef Expression
1 bnj1405.1 . 2  |-  ( ph  ->  X  e.  U_ y  e.  A  B )
2 eliun 4298 . 2  |-  ( X  e.  U_ y  e.  A  B  <->  E. y  e.  A  X  e.  B )
31, 2sylib 199 1  |-  ( ph  ->  E. y  e.  A  X  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1867   E.wrex 2774   U_ciun 4293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rex 2779  df-v 3080  df-iun 4295
This theorem is referenced by:  bnj1408  29674  bnj1450  29688  bnj1501  29705
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