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Theorem sbccom2lem 30497
Description: Lemma for sbccom2 30498. (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 sbcan 3354 . . . 4  |-  ( [. A  /  x ]. (
y  =  B  /\  ph )  <->  ( [. A  /  x ]. y  =  B  /\  [. A  /  x ]. ph )
)
2 sbc5 3336 . . . 4  |-  ( [. A  /  x ]. (
y  =  B  /\  ph )  <->  E. x ( x  =  A  /\  (
y  =  B  /\  ph ) ) )
3 sbccom2lem.1 . . . . . 6  |-  A  e. 
_V
43csbconstgi 30490 . . . . . 6  |-  [_ A  /  x ]_ y  =  y
5 eqid 2441 . . . . . 6  |-  [_ A  /  x ]_ B  = 
[_ A  /  x ]_ B
63, 4, 5sbceqi 30481 . . . . 5  |-  ( [. A  /  x ]. y  =  B  <->  y  =  [_ A  /  x ]_ B
)
76anbi1i 695 . . . 4  |-  ( (
[. A  /  x ]. y  =  B  /\  [. A  /  x ]. ph )  <->  ( y  =  [_ A  /  x ]_ B  /\  [. A  /  x ]. ph )
)
81, 2, 73bitr3i 275 . . 3  |-  ( E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  ( y  =  [_ A  /  x ]_ B  /\  [. A  /  x ]. ph )
)
98exbii 1652 . 2  |-  ( E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y ( y  = 
[_ A  /  x ]_ B  /\  [. A  /  x ]. ph )
)
10 sbc5 3336 . . . . 5  |-  ( [. B  /  y ]. ph  <->  E. y
( y  =  B  /\  ph ) )
1110sbcbii 3371 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. E. y ( y  =  B  /\  ph ) )
12 sbc5 3336 . . . 4  |-  ( [. A  /  x ]. E. y ( y  =  B  /\  ph )  <->  E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
1311, 12bitri 249 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  E. x
( x  =  A  /\  E. y ( y  =  B  /\  ph ) ) )
14 19.42v 1759 . . . . . 6  |-  ( E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  ( x  =  A  /\  E. y
( y  =  B  /\  ph ) ) )
1514bicomi 202 . . . . 5  |-  ( ( x  =  A  /\  E. y ( y  =  B  /\  ph )
)  <->  E. y ( x  =  A  /\  (
y  =  B  /\  ph ) ) )
1615exbii 1652 . . . 4  |-  ( E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) )  <->  E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
17 excom 1833 . . . 4  |-  ( E. x E. y ( x  =  A  /\  ( y  =  B  /\  ph ) )  <->  E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
1816, 17bitri 249 . . 3  |-  ( E. x ( x  =  A  /\  E. y
( y  =  B  /\  ph ) )  <->  E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
1913, 18bitri 249 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  E. y E. x ( x  =  A  /\  ( y  =  B  /\  ph ) ) )
20 sbc5 3336 . 2  |-  ( [. [_ A  /  x ]_ B  /  y ]. [. A  /  x ]. ph  <->  E. y
( y  =  [_ A  /  x ]_ B  /\  [. A  /  x ]. ph ) )
219, 19, 203bitr4i 277 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 1381   E.wex 1597    e. wcel 1802   _Vcvv 3093   [.wsbc 3311   [_csb 3417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-v 3095  df-sbc 3312  df-csb 3418
This theorem is referenced by:  sbccom2  30498
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