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Theorem sbccom2 30506
Description: Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypothesis
Ref Expression
sbccom2.1  |-  A  e. 
_V
Assertion
Ref Expression
sbccom2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. [. A  /  x ]. ph )
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    A( x)    B( x, y)

Proof of Theorem sbccom2
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcco 3336 . . . . . . 7  |-  ( [. B  /  w ]. [. w  /  y ]. ph  <->  [. B  / 
y ]. ph )
21bicomi 202 . . . . . 6  |-  ( [. B  /  y ]. ph  <->  [. B  /  w ]. [. w  / 
y ]. ph )
32sbcbii 3373 . . . . 5  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
4 sbcco 3336 . . . . . 6  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
54bicomi 202 . . . . 5  |-  ( [. A  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  / 
z ]. [. z  /  x ]. [. B  /  w ]. [. w  / 
y ]. ph )
6 vex 3098 . . . . . . 7  |-  z  e. 
_V
76sbccom2lem 30505 . . . . . 6  |-  ( [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph )
87sbcbii 3373 . . . . 5  |-  ( [. A  /  z ]. [. z  /  x ]. [. B  /  w ]. [. w  /  y ]. ph  <->  [. A  / 
z ]. [. [_ z  /  x ]_ B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph )
93, 5, 83bitri 271 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. A  / 
z ]. [. [_ z  /  x ]_ B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph )
10 sbccom2.1 . . . . 5  |-  A  e. 
_V
1110sbccom2lem 30505 . . . 4  |-  ( [. A  /  z ]. [. [_ z  /  x ]_ B  /  w ]. [. z  /  x ]. [. w  / 
y ]. ph  <->  [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  / 
z ]. [. z  /  x ]. [. w  / 
y ]. ph )
12 sbcco 3336 . . . . 5  |-  ( [. A  /  z ]. [. z  /  x ]. [. w  /  y ]. ph  <->  [. A  /  x ]. [. w  / 
y ]. ph )
1312sbcbii 3373 . . . 4  |-  ( [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  /  z ]. [. z  /  x ]. [. w  /  y ]. ph  <->  [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  / 
y ]. ph )
149, 11, 133bitri 271 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  / 
y ]. ph )
15 csbco 3430 . . . 4  |-  [_ A  /  z ]_ [_ z  /  x ]_ B  = 
[_ A  /  x ]_ B
16 dfsbcq 3315 . . . 4  |-  ( [_ A  /  z ]_ [_ z  /  x ]_ B  = 
[_ A  /  x ]_ B  ->  ( [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  / 
y ]. ph ) )
1715, 16ax-mp 5 . . 3  |-  ( [. [_ A  /  z ]_ [_ z  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  / 
y ]. ph )
1814, 17bitri 249 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  / 
y ]. ph )
19 sbccom 3393 . . 3  |-  ( [. A  /  x ]. [. w  /  y ]. ph  <->  [. w  / 
y ]. [. A  /  x ]. ph )
2019sbcbii 3373 . 2  |-  ( [. [_ A  /  x ]_ B  /  w ]. [. A  /  x ]. [. w  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  w ]. [. w  / 
y ]. [. A  /  x ]. ph )
21 sbcco 3336 . 2  |-  ( [. [_ A  /  x ]_ B  /  w ]. [. w  /  y ]. [. A  /  x ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. [. A  /  x ]. ph )
2218, 20, 213bitri 271 1  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1383    e. wcel 1804   _Vcvv 3095   [.wsbc 3313   [_csb 3420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-sbc 3314  df-csb 3421
This theorem is referenced by:  sbccom2f  30507
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