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Theorem sbccom2 30121
Description: Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypothesis
Ref Expression
sbccom2.1  |-  A  e. 
_V
Assertion
Ref Expression
sbccom2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. [. A  /  x ]. ph )
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    A( x)    B( x, y)

Proof of Theorem sbccom2
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcco 3347 . . . . . . . 8  |-  ( [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. ph )
21bicomi 202 . . . . . . 7  |-  ( [. B  /  y ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. ph )
32sbcbii 3384 . . . . . 6  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
4 sbcco 3347 . . . . . . 7  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
54bicomi 202 . . . . . 6  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  / 
z ]. [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
63, 5bitri 249 . . . . 5  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  / 
z ]. [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
7 vex 3109 . . . . . . 7  |-  z  e. 
_V
87sbccom2lem 30120 . . . . . 6  |-  ( [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph )
98sbcbii 3384 . . . . 5  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  / 
z ]. [. [_ z  /  x ]_ B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph )
106, 9bitri 249 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  / 
z ]. [. [_ z  /  x ]_ B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph )
11 sbccom2.1 . . . . 5  |-  A  e. 
_V
1211sbccom2lem 30120 . . . 4  |-  ( [. A  /  z ]. [. [_ z  /  x ]_ B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph  <->  [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  / 
z ]. [. z  /  x ]. [. w  / 
y ]. ph )
13 sbcco 3347 . . . . 5  |-  ( [. A  /  z ]. [. z  /  x ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. w  / 
y ]. ph )
1413sbcbii 3384 . . . 4  |-  ( [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  /  z ]. [. z  /  x ]. [. w  /  y ]. ph  <->  [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  / 
y ]. ph )
1510, 12, 143bitri 271 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  / 
y ]. ph )
16 csbco 3438 . . . 4  |-  [_ A  /  z ]_ [_ z  /  x ]_ B  = 
[_ A  /  x ]_ B
17 dfsbcq 3326 . . . 4  |-  ( [_ A  /  z ]_ [_ z  /  x ]_ B  = 
[_ A  /  x ]_ B  ->  ( [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  / 
y ]. ph ) )
1816, 17ax-mp 5 . . 3  |-  ( [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  / 
y ]. ph )
1915, 18bitri 249 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  / 
y ]. ph )
20 sbccom 3404 . . 3  |-  ( [. A  /  x ]. [. w  /  y ]. ph  <->  [. w  / 
y ]. [. A  /  x ]. ph )
2120sbcbii 3384 . 2  |-  ( [. [_ A  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  w ]. [. w  / 
y ]. [. A  /  x ]. ph )
22 sbcco 3347 . 2  |-  ( [. [_ A  /  x ]_ B  /  w ]. [. w  /  y ]. [. A  /  x ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. [. A  /  x ]. ph )
2319, 21, 223bitri 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 1374    e. wcel 1762   _Vcvv 3106   [.wsbc 3324   [_csb 3428
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 1961  ax-ext 2438
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 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-sbc 3325  df-csb 3429
This theorem is referenced by:  sbccom2f  30122
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