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Theorem rexcom4b 2809
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 2808 . 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 2794 . . . 4  |-  E. x  x  =  B
43biantru 491 . . 3  |-  ( ph  <->  (
ph  /\  E. x  x  =  B )
)
54rexbii 2568 . 2  |-  ( E. y  e.  A  ph  <->  E. y  e.  A  (
ph  /\  E. x  x  =  B )
)
61, 5bitr4i 243 1  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788
This theorem is referenced by:  rexcom4bOLD  26345  islshpat  29207
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-rex 2549  df-v 2790
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