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Theorem sbccom2lem 30122
Description: Lemma for sbccom2 30123. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypothesis
Ref Expression
sbccom2lem.1  |-  A  e. 
_V
Assertion
Ref Expression
sbccom2lem  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. [. A  /  x ]. ph )
Distinct variable groups:    x, y, A    y, B
Allowed substitution hints:    ph( x, y)    B( x)

Proof of Theorem sbccom2lem
StepHypRef Expression
1 sbc5 3351 . . . . . 6  |-  ( [. B  /  y ]. ph  <->  E. y
( y  =  B  /\  ph ) )
21sbcbii 3386 . . . . 5  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. E. y ( y  =  B  /\  ph ) )
3 sbc5 3351 . . . . 5  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
42, 3bitri 249 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  E. x
( x  =  A  /\  E. y ( y  =  B  /\  ph ) ) )
5 19.42v 1944 . . . . . . 7  |-  ( E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
65bicomi 202 . . . . . 6  |-  ( ( x  =  A  /\  E. y ( y  =  B  /\  ph )
)  <->  E. y ( x  =  A  /\  (
y  =  B  /\  ph ) ) )
76exbii 1639 . . . . 5  |-  ( E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) )  <->  E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
8 excom 1793 . . . . 5  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
97, 8bitri 249 . . . 4  |-  ( E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) )  <->  E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
104, 9bitri 249 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
11 sbc5 3351 . . . . 5  |-  ( [. A  /  x ]. (
y  =  B  /\  ph )  <->  E. x ( x  =  A  /\  (
y  =  B  /\  ph ) ) )
12 sbcan 3369 . . . . . 6  |-  ( [. A  /  x ]. (
y  =  B  /\  ph )  <->  ( [. A  /  x ]. y  =  B  /\  [. A  /  x ]. ph )
)
13 sbccom2lem.1 . . . . . . . 8  |-  A  e. 
_V
1413csbconstgi 30114 . . . . . . . 8  |-  [_ A  /  x ]_ y  =  y
15 eqid 2462 . . . . . . . 8  |-  [_ A  /  x ]_ B  = 
[_ A  /  x ]_ B
1613, 14, 15sbceqi 30105 . . . . . . 7  |-  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
)
1716anbi1i 695 . . . . . 6  |-  ( (
[. A  /  x ]. y  =  B  /\  [. A  /  x ]. ph )  <->  ( y  =  [_ A  /  x ]_ B  /\  [. A  /  x ]. ph )
)
1812, 17bitri 249 . . . . 5  |-  ( [. A  /  x ]. (
y  =  B  /\  ph )  <->  ( y  = 
[_ A  /  x ]_ B  /\  [. A  /  x ]. ph )
)
1911, 18bitr3i 251 . . . 4  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  ( y  =  [_ A  /  x ]_ B  /\  [. A  /  x ]. ph )
)
2019exbii 1639 . . 3  |-  ( E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y ( y  = 
[_ A  /  x ]_ B  /\  [. A  /  x ]. ph )
)
2110, 20bitri 249 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  E. y
( y  =  [_ A  /  x ]_ B  /\  [. A  /  x ]. ph ) )
22 sbc5 3351 . 2  |-  ( [. [_ A  /  x ]_ B  /  y ]. [. A  /  x ]. ph  <->  E. y
( y  =  [_ A  /  x ]_ B  /\  [. A  /  x ]. ph ) )
2321, 22bitr4i 252 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762   _Vcvv 3108   [.wsbc 3326   [_csb 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-v 3110  df-sbc 3327  df-csb 3431
This theorem is referenced by:  sbccom2  30123
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