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Theorem sbccomlem 3247
Description: Lemma for sbccom 3248. (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 1759 . . . 4  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
2 exdistr 1933 . . . 4  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
3 an12 774 . . . . . . 7  |-  ( ( x  =  A  /\  ( y  =  B  /\  ph ) )  <-> 
( y  =  B  /\  ( x  =  A  /\  ph )
) )
43exbii 1593 . . . . . 6  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. x
( y  =  B  /\  ( x  =  A  /\  ph )
) )
5 19.42v 1932 . . . . . 6  |-  ( E. x ( y  =  B  /\  ( x  =  A  /\  ph ) )  <->  ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
64, 5bitri 242 . . . . 5  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
76exbii 1593 . . . 4  |-  ( E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
81, 2, 73bitr3i 268 . . 3  |-  ( E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
9 sbc5 3194 . . 3  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
10 sbc5 3194 . . 3  |-  ( [. B  /  y ]. E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  B  /\  E. x
( x  =  A  /\  ph ) ) )
118, 9, 103bitr4i 270 . 2  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  [. B  /  y ]. E. x ( x  =  A  /\  ph )
)
12 sbc5 3194 . . 3  |-  ( [. B  /  y ]. ph  <->  E. y
( y  =  B  /\  ph ) )
1312sbcbii 3228 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. E. y ( y  =  B  /\  ph ) )
14 sbc5 3194 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
1514sbcbii 3228 . 2  |-  ( [. B  /  y ]. [. A  /  x ]. ph  <->  [. B  / 
y ]. E. x ( x  =  A  /\  ph ) )
1611, 13, 153bitr4i 270 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1654   [.wsbc 3170
This theorem is referenced by:  sbccom  3248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2967  df-sbc 3171
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