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Theorem sbccom 3233
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 3232 . . . 4  |-  ( [. A  /  z ]. [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. A  / 
z ]. [. w  / 
y ]. [. z  /  x ]. ph )
2 sbccomlem 3232 . . . . . . 7  |-  ( [. w  /  y ]. [. z  /  x ]. ph  <->  [. z  /  x ]. [. w  / 
y ]. ph )
32sbcbii 3217 . . . . . 6  |-  ( [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph )
4 sbccomlem 3232 . . . . . 6  |-  ( [. B  /  w ]. [. z  /  x ]. [. w  /  y ]. ph  <->  [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
53, 4bitri 242 . . . . 5  |-  ( [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
65sbcbii 3217 . . . 4  |-  ( [. A  /  z ]. [. B  /  w ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. A  / 
z ]. [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
7 sbccomlem 3232 . . . . 5  |-  ( [. A  /  z ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
87sbcbii 3217 . . . 4  |-  ( [. B  /  w ]. [. A  /  z ]. [. w  /  y ]. [. z  /  x ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
91, 6, 83bitr3i 268 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
10 sbcco 3184 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
11 sbcco 3184 . . 3  |-  ( [. B  /  w ]. [. w  /  y ]. [. A  /  z ]. [. z  /  x ]. ph  <->  [. B  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
129, 10, 113bitr3i 268 . 2  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. [. A  / 
z ]. [. z  /  x ]. ph )
13 sbcco 3184 . . 3  |-  ( [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. ph )
1413sbcbii 3217 . 2  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  / 
y ]. ph )
15 sbcco 3184 . . 3  |-  ( [. A  /  z ]. [. z  /  x ]. ph  <->  [. A  /  x ]. ph )
1615sbcbii 3217 . 2  |-  ( [. B  /  y ]. [. A  /  z ]. [. z  /  x ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
1712, 14, 163bitr3i 268 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   [.wsbc 3162
This theorem is referenced by:  csbcomg  3275  csbabg  3311  mpt2xopovel  6472  elmptrab  17860  sbcrot3  26848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-sbc 3163
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