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Theorem bnj1405 33191
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 4330 . 2  |-  ( X  e.  U_ y  e.  A  B  <->  E. y  e.  A  X  e.  B )
31, 2sylib 196 1  |-  ( ph  ->  E. y  e.  A  X  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   E.wrex 2815   U_ciun 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-iun 4327
This theorem is referenced by:  bnj1408  33388  bnj1450  33402  bnj1501  33419
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