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Theorem sbccom 3411
Description: Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
Assertion
Ref Expression
sbccom  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem sbccom
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbccomlem 3410 . . . 4  |-  ( [. A  /  z ]. [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. A  / 
z ]. [. w  / 
y ]. [. z  /  x ]. ph )
2 sbccomlem 3410 . . . . . . 7  |-  ( [. w  /  y ]. [. z  /  x ]. ph  <->  [. z  /  x ]. [. w  / 
y ]. ph )
32sbcbii 3391 . . . . . 6  |-  ( [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph )
4 sbccomlem 3410 . . . . . 6  |-  ( [. B  /  w ]. [. z  /  x ]. [. w  /  y ]. ph  <->  [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
53, 4bitri 249 . . . . 5  |-  ( [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
65sbcbii 3391 . . . 4  |-  ( [. A  /  z ]. [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. A  / 
z ]. [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
7 sbccomlem 3410 . . . . 5  |-  ( [. A  /  z ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
87sbcbii 3391 . . . 4  |-  ( [. B  /  w ]. [. A  /  z ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
91, 6, 83bitr3i 275 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
10 sbcco 3354 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
11 sbcco 3354 . . 3  |-  ( [. B  /  w ]. [. w  /  y ]. [. A  /  z ]. [. z  /  x ]. ph  <->  [. B  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
129, 10, 113bitr3i 275 . 2  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
13 sbcco 3354 . . 3  |-  ( [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. ph )
1413sbcbii 3391 . 2  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  / 
y ]. ph )
15 sbcco 3354 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. ph  <->  [. A  /  x ]. ph )
1615sbcbii 3391 . 2  |-  ( [. B  /  y ]. [. A  /  z ]. [. z  /  x ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
1712, 14, 163bitr3i 275 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   [.wsbc 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115  df-sbc 3332
This theorem is referenced by:  csbcom  3837  csbcomgOLD  3838  csbab  3855  csbabgOLD  3856  mpt2xopovel  6949  wrd2ind  12669  elmptrab  20155  sbccom2  30361  sbcrot3  30555
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