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Theorem bnj1405 28869
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 3909 . 2  |-  ( X  e.  U_ y  e.  A  B  <->  E. y  e.  A  X  e.  B )
31, 2sylib 188 1  |-  ( ph  ->  E. y  e.  A  X  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   E.wrex 2544   U_ciun 3905
This theorem is referenced by:  bnj1408  29066  bnj1450  29080  bnj1501  29097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-iun 3907
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