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Theorem sbccom 3062
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 3061 . . . 4  |-  ( [. A  /  z ]. [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. A  / 
z ]. [. w  / 
y ]. [. z  /  x ]. ph )
2 sbccomlem 3061 . . . . . . 7  |-  ( [. w  /  y ]. [. z  /  x ]. ph  <->  [. z  /  x ]. [. w  / 
y ]. ph )
32sbcbii 3046 . . . . . 6  |-  ( [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph )
4 sbccomlem 3061 . . . . . 6  |-  ( [. B  /  w ]. [. z  /  x ]. [. w  /  y ]. ph  <->  [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
53, 4bitri 240 . . . . 5  |-  ( [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
65sbcbii 3046 . . . 4  |-  ( [. A  /  z ]. [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. A  / 
z ]. [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
7 sbccomlem 3061 . . . . 5  |-  ( [. A  /  z ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
87sbcbii 3046 . . . 4  |-  ( [. B  /  w ]. [. A  /  z ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
91, 6, 83bitr3i 266 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
10 sbcco 3013 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
11 sbcco 3013 . . 3  |-  ( [. B  /  w ]. [. w  /  y ]. [. A  /  z ]. [. z  /  x ]. ph  <->  [. B  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
129, 10, 113bitr3i 266 . 2  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
13 sbcco 3013 . . 3  |-  ( [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. ph )
1413sbcbii 3046 . 2  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  / 
y ]. ph )
15 sbcco 3013 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. ph  <->  [. A  /  x ]. ph )
1615sbcbii 3046 . 2  |-  ( [. B  /  y ]. [. A  /  z ]. [. z  /  x ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
1712, 14, 163bitr3i 266 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [.wsbc 2991
This theorem is referenced by:  csbcomg  3104  csbabg  3142  elmptrab  17522  sbcrot3  26868  mpt2xopovel  28086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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