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Theorem sbccomlem 3037
Description: Lemma for sbccom 3038. (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 1664 . . . 4  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
2 exdistr 1909 . . . 4  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
3 an12 561 . . . . . . 7  |-  ( ( x  =  A  /\  ( y  =  B  /\  ph ) )  <-> 
( y  =  B  /\  ( x  =  A  /\  ph )
) )
43exbii 1605 . . . . . 6  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. x
( y  =  B  /\  ( x  =  A  /\  ph )
) )
5 19.42v 1906 . . . . . 6  |-  ( E. x ( y  =  B  /\  ( x  =  A  /\  ph ) )  <->  ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
64, 5bitri 184 . . . . 5  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
76exbii 1605 . . . 4  |-  ( E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
81, 2, 73bitr3i 210 . . 3  |-  ( E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
9 sbc5 2986 . . 3  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
10 sbc5 2986 . . 3  |-  ( [. B  /  y ]. E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
118, 9, 103bitr4i 212 . 2  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  [. B  /  y ]. E. x ( x  =  A  /\  ph )
)
12 sbc5 2986 . . 3  |-  ( [. B  /  y ]. ph  <->  E. y
( y  =  B  /\  ph ) )
1312sbcbii 3022 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. E. y ( y  =  B  /\  ph ) )
14 sbc5 2986 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
1514sbcbii 3022 . 2  |-  ( [. B  /  y ]. [. A  /  x ]. ph  <->  [. B  / 
y ]. E. x ( x  =  A  /\  ph ) )
1611, 13, 153bitr4i 212 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353   E.wex 1492   [.wsbc 2962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963
This theorem is referenced by:  sbccom  3038
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