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Theorem rexcom4b 2941
Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
Hypothesis
Ref Expression
rexcom4b.1  |-  B  e. 
_V
Assertion
Ref Expression
rexcom4b  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ph )
Distinct variable groups:    x, A    x, y    ph, x    x, B
Allowed substitution hints:    ph( y)    A( y)    B( y)

Proof of Theorem rexcom4b
StepHypRef Expression
1 rexcom4a 2940 . 2  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  (
ph  /\  E. x  x  =  B )
)
2 rexcom4b.1 . . . . 5  |-  B  e. 
_V
32isseti 2926 . . . 4  |-  E. x  x  =  B
43biantru 492 . . 3  |-  ( ph  <->  (
ph  /\  E. x  x  =  B )
)
54rexbii 2695 . 2  |-  ( E. y  e.  A  ph  <->  E. y  e.  A  (
ph  /\  E. x  x  =  B )
)
61, 5bitr4i 244 1  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   E.wrex 2671   _Vcvv 2920
This theorem is referenced by:  islshpat  29504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rex 2676  df-v 2922
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