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Theorem sbccom 3371
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 3370 . . . 4  |-  ( [. A  /  z ]. [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. A  / 
z ]. [. w  / 
y ]. [. z  /  x ]. ph )
2 sbccomlem 3370 . . . . . . 7  |-  ( [. w  /  y ]. [. z  /  x ]. ph  <->  [. z  /  x ]. [. w  / 
y ]. ph )
32sbcbii 3355 . . . . . 6  |-  ( [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph )
4 sbccomlem 3370 . . . . . 6  |-  ( [. B  /  w ]. [. z  /  x ]. [. w  /  y ]. ph  <->  [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
53, 4bitri 252 . . . . 5  |-  ( [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
65sbcbii 3355 . . . 4  |-  ( [. A  /  z ]. [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. A  / 
z ]. [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
7 sbccomlem 3370 . . . . 5  |-  ( [. A  /  z ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
87sbcbii 3355 . . . 4  |-  ( [. B  /  w ]. [. A  /  z ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
91, 6, 83bitr3i 278 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
10 sbcco 3322 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
11 sbcco 3322 . . 3  |-  ( [. B  /  w ]. [. w  /  y ]. [. A  /  z ]. [. z  /  x ]. ph  <->  [. B  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
129, 10, 113bitr3i 278 . 2  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
13 sbcco 3322 . . 3  |-  ( [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. ph )
1413sbcbii 3355 . 2  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  / 
y ]. ph )
15 sbcco 3322 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. ph  <->  [. A  /  x ]. ph )
1615sbcbii 3355 . 2  |-  ( [. B  /  y ]. [. A  /  z ]. [. z  /  x ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
1712, 14, 163bitr3i 278 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   [.wsbc 3299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-v 3083  df-sbc 3300
This theorem is referenced by:  csbcom  3811  csbab  3825  mpt2xopovel  6970  wrd2ind  12824  elmptrab  20828  sbccom2  32278  sbcrot3  35552  csbabgOLD  37071
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