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Theorem bnj1405 29185
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 3925 . 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 1696   E.wrex 2557   U_ciun 3921
This theorem is referenced by:  bnj1408  29382  bnj1450  29396  bnj1501  29413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-iun 3923
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