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Theorem sbccom2f 30499
Description: Commutative law for double class substitution, with non free variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypotheses
Ref Expression
sbccom2f.1  |-  A  e. 
_V
sbccom2f.2  |-  F/_ y A
Assertion
Ref Expression
sbccom2f  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. [. A  /  x ]. ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)    B( x, y)

Proof of Theorem sbccom2f
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbcco 3334 . . . 4  |-  ( [. B  /  z ]. [. z  /  y ]. ph  <->  [. B  / 
y ]. ph )
21bicomi 202 . . 3  |-  ( [. B  /  y ]. ph  <->  [. B  / 
z ]. [. z  / 
y ]. ph )
32sbcbii 3371 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. [. B  / 
z ]. [. z  / 
y ]. ph )
4 sbccom2f.1 . . 3  |-  A  e. 
_V
54sbccom2 30498 . 2  |-  ( [. A  /  x ]. [. B  /  z ]. [. z  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
z ]. [. A  /  x ]. [. z  / 
y ]. ph )
6 vex 3096 . . . . . . 7  |-  z  e. 
_V
76sbccom2 30498 . . . . . 6  |-  ( [. z  /  y ]. [. A  /  x ]. ph  <->  [. [_ z  /  y ]_ A  /  x ]. [. z  /  y ]. ph )
8 sbccom2f.2 . . . . . . . 8  |-  F/_ y A
9 eqidd 2442 . . . . . . . 8  |-  ( y  =  z  ->  A  =  A )
106, 8, 9csbief 3442 . . . . . . 7  |-  [_ z  /  y ]_ A  =  A
11 dfsbcq 3313 . . . . . . 7  |-  ( [_ z  /  y ]_ A  =  A  ->  ( [. [_ z  /  y ]_ A  /  x ]. [. z  /  y ]. ph  <->  [. A  /  x ]. [. z  / 
y ]. ph ) )
1210, 11ax-mp 5 . . . . . 6  |-  ( [. [_ z  /  y ]_ A  /  x ]. [. z  /  y ]. ph  <->  [. A  /  x ]. [. z  / 
y ]. ph )
137, 12bitri 249 . . . . 5  |-  ( [. z  /  y ]. [. A  /  x ]. ph  <->  [. A  /  x ]. [. z  / 
y ]. ph )
1413bicomi 202 . . . 4  |-  ( [. A  /  x ]. [. z  /  y ]. ph  <->  [. z  / 
y ]. [. A  /  x ]. ph )
1514sbcbii 3371 . . 3  |-  ( [. [_ A  /  x ]_ B  /  z ]. [. A  /  x ]. [. z  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
z ]. [. z  / 
y ]. [. A  /  x ]. ph )
16 sbcco 3334 . . 3  |-  ( [. [_ A  /  x ]_ B  /  z ]. [. z  /  y ]. [. A  /  x ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. [. A  /  x ]. ph )
1715, 16bitri 249 . 2  |-  ( [. [_ A  /  x ]_ B  /  z ]. [. A  /  x ]. [. z  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. [. A  /  x ]. ph )
183, 5, 173bitri 271 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1381    e. wcel 1802   F/_wnfc 2589   _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-3an 974  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:  sbccom2fi  30500
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