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Theorem sbccomlem 3199
Description: Lemma for sbccom 3200. (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 1752 . . . 4  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
2 exdistr 1925 . . . 4  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
3 an12 773 . . . . . . 7  |-  ( ( x  =  A  /\  ( y  =  B  /\  ph ) )  <-> 
( y  =  B  /\  ( x  =  A  /\  ph )
) )
43exbii 1589 . . . . . 6  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. x
( y  =  B  /\  ( x  =  A  /\  ph )
) )
5 19.42v 1924 . . . . . 6  |-  ( E. x ( y  =  B  /\  ( x  =  A  /\  ph ) )  <->  ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
64, 5bitri 241 . . . . 5  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
76exbii 1589 . . . 4  |-  ( E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
81, 2, 73bitr3i 267 . . 3  |-  ( E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
9 sbc5 3153 . . 3  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
10 sbc5 3153 . . 3  |-  ( [. B  /  y ]. E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
118, 9, 103bitr4i 269 . 2  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  [. B  /  y ]. E. x ( x  =  A  /\  ph )
)
12 sbc5 3153 . . 3  |-  ( [. B  /  y ]. ph  <->  E. y
( y  =  B  /\  ph ) )
1312sbcbii 3184 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. E. y ( y  =  B  /\  ph ) )
14 sbc5 3153 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
1514sbcbii 3184 . 2  |-  ( [. B  /  y ]. [. A  /  x ]. ph  <->  [. B  / 
y ]. E. x ( x  =  A  /\  ph ) )
1611, 13, 153bitr4i 269 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649   [.wsbc 3129
This theorem is referenced by:  sbccom  3200
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 2393
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 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-sbc 3130
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