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Theorem sbccom 3040
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 3039 . . . 4  |-  ( [. A  /  z ]. [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. A  / 
z ]. [. w  / 
y ]. [. z  /  x ]. ph )
2 sbccomlem 3039 . . . . . . 7  |-  ( [. w  /  y ]. [. z  /  x ]. ph  <->  [. z  /  x ]. [. w  / 
y ]. ph )
32sbcbii 3024 . . . . . 6  |-  ( [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph )
4 sbccomlem 3039 . . . . . 6  |-  ( [. B  /  w ]. [. z  /  x ]. [. w  /  y ]. ph  <->  [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
53, 4bitri 184 . . . . 5  |-  ( [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
65sbcbii 3024 . . . 4  |-  ( [. A  /  z ]. [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. A  / 
z ]. [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
7 sbccomlem 3039 . . . . 5  |-  ( [. A  /  z ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
87sbcbii 3024 . . . 4  |-  ( [. B  /  w ]. [. A  /  z ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
91, 6, 83bitr3i 210 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
10 sbcco 2986 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
11 sbcco 2986 . . 3  |-  ( [. B  /  w ]. [. w  /  y ]. [. A  /  z ]. [. z  /  x ]. ph  <->  [. B  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
129, 10, 113bitr3i 210 . 2  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
13 sbcco 2986 . . 3  |-  ( [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. ph )
1413sbcbii 3024 . 2  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  / 
y ]. ph )
15 sbcco 2986 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. ph  <->  [. A  /  x ]. ph )
1615sbcbii 3024 . 2  |-  ( [. B  /  y ]. [. A  /  z ]. [. z  /  x ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
1712, 14, 163bitr3i 210 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   [.wsbc 2964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965
This theorem is referenced by:  csbcomg  3082  csbabg  3120  mpoxopovel  6244
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