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Theorem sbccomlem 2913
Description: Lemma for sbccom 2914. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
Assertion
Ref Expression
sbccomlem  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Distinct variable groups:    x, y, A   
x, B, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sbccomlem
StepHypRef Expression
1 excom 1599 . . . 4  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
2 exdistr 1835 . . . 4  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
3 an12 528 . . . . . . 7  |-  ( ( x  =  A  /\  ( y  =  B  /\  ph ) )  <-> 
( y  =  B  /\  ( x  =  A  /\  ph )
) )
43exbii 1541 . . . . . 6  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. x
( y  =  B  /\  ( x  =  A  /\  ph )
) )
5 19.42v 1834 . . . . . 6  |-  ( E. x ( y  =  B  /\  ( x  =  A  /\  ph ) )  <->  ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
64, 5bitri 182 . . . . 5  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
76exbii 1541 . . . 4  |-  ( E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
81, 2, 73bitr3i 208 . . 3  |-  ( E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
9 sbc5 2863 . . 3  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
10 sbc5 2863 . . 3  |-  ( [. B  /  y ]. E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
118, 9, 103bitr4i 210 . 2  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  [. B  /  y ]. E. x ( x  =  A  /\  ph )
)
12 sbc5 2863 . . 3  |-  ( [. B  /  y ]. ph  <->  E. y
( y  =  B  /\  ph ) )
1312sbcbii 2898 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. E. y ( y  =  B  /\  ph ) )
14 sbc5 2863 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
1514sbcbii 2898 . 2  |-  ( [. B  /  y ]. [. A  /  x ]. ph  <->  [. B  / 
y ]. E. x ( x  =  A  /\  ph ) )
1611, 13, 153bitr4i 210 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426   [.wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841
This theorem is referenced by:  sbccom  2914
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