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Theorem sbccomlem 3376
Description: Lemma for sbccom 3377. (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 1901 . . . 4  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
2 exdistr 1827 . . . 4  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
3 an12 804 . . . . . . 7  |-  ( ( x  =  A  /\  ( y  =  B  /\  ph ) )  <-> 
( y  =  B  /\  ( x  =  A  /\  ph )
) )
43exbii 1714 . . . . . 6  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. x
( y  =  B  /\  ( x  =  A  /\  ph )
) )
5 19.42v 1826 . . . . . 6  |-  ( E. x ( y  =  B  /\  ( x  =  A  /\  ph ) )  <->  ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
64, 5bitri 252 . . . . 5  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
76exbii 1714 . . . 4  |-  ( E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
81, 2, 73bitr3i 278 . . 3  |-  ( E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
9 sbc5 3330 . . 3  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
10 sbc5 3330 . . 3  |-  ( [. B  /  y ]. E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
118, 9, 103bitr4i 280 . 2  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  [. B  /  y ]. E. x ( x  =  A  /\  ph )
)
12 sbc5 3330 . . 3  |-  ( [. B  /  y ]. ph  <->  E. y
( y  =  B  /\  ph ) )
1312sbcbii 3361 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. E. y ( y  =  B  /\  ph ) )
14 sbc5 3330 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
1514sbcbii 3361 . 2  |-  ( [. B  /  y ]. [. A  /  x ]. ph  <->  [. B  / 
y ]. E. x ( x  =  A  /\  ph ) )
1611, 13, 153bitr4i 280 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659   [.wsbc 3305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3089  df-sbc 3306
This theorem is referenced by:  sbccom  3377
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