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Theorem csbcom 3796
Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
csbcom  |-  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_ B  /  y ]_ [_ A  /  x ]_ C
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    A( x)    B( y)    C( x, y)

Proof of Theorem csbcom
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbccom 3372 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. z  e.  C  <->  [. B  /  y ]. [. A  /  x ]. z  e.  C
)
2 sbcel2 3790 . . . . 5  |-  ( [. B  /  y ]. z  e.  C  <->  z  e.  [_ B  /  y ]_ C
)
32sbcbii 3352 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. z  e.  C  <->  [. A  /  x ]. z  e.  [_ B  /  y ]_ C
)
4 sbcel2 3790 . . . . 5  |-  ( [. A  /  x ]. z  e.  C  <->  z  e.  [_ A  /  x ]_ C
)
54sbcbii 3352 . . . 4  |-  ( [. B  /  y ]. [. A  /  x ]. z  e.  C  <->  [. B  /  y ]. z  e.  [_ A  /  x ]_ C )
61, 3, 53bitr3i 275 . . 3  |-  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  [. B  /  y ]. z  e.  [_ A  /  x ]_ C )
7 sbcel2 3790 . . 3  |-  ( [. A  /  x ]. z  e.  [_ B  /  y ]_ C  <->  z  e.  [_ A  /  x ]_ [_ B  /  y ]_ C
)
8 sbcel2 3790 . . 3  |-  ( [. B  /  y ]. z  e.  [_ A  /  x ]_ C  <->  z  e.  [_ B  /  y ]_ [_ A  /  x ]_ C )
96, 7, 83bitr3i 275 . 2  |-  ( z  e.  [_ A  /  x ]_ [_ B  / 
y ]_ C  <->  z  e.  [_ B  /  y ]_ [_ A  /  x ]_ C )
109eqriv 2450 1  |-  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_ B  /  y ]_ [_ A  /  x ]_ C
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   [.wsbc 3292   [_csb 3394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-in 3442  df-ss 3449  df-nul 3745
This theorem is referenced by:  ovmpt2s  6323  fvmpt2curryd  6899
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