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Theorem csbmpt2 4726
Description: Move substitution into the second part of a maps-to notation. (Contributed by AV, 26-Sep-2019.)
Assertion
Ref Expression
csbmpt2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( y  e.  Y  |->  Z )  =  ( y  e.  Y  |->  [_ A  /  x ]_ Z ) )
Distinct variable groups:    y, A    y, V    y, Y, x
Allowed substitution hints:    A( x)    V( x)    Z( x, y)

Proof of Theorem csbmpt2
StepHypRef Expression
1 csbmpt12 4725 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( y  e.  Y  |->  Z )  =  ( y  e. 
[_ A  /  x ]_ Y  |->  [_ A  /  x ]_ Z ) )
2 csbconstg 3403 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ Y  =  Y )
32mpteq1d 4476 . 2  |-  ( A  e.  V  ->  (
y  e.  [_ A  /  x ]_ Y  |->  [_ A  /  x ]_ Z
)  =  ( y  e.  Y  |->  [_ A  /  x ]_ Z ) )
41, 3eqtrd 2493 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( y  e.  Y  |->  Z )  =  ( y  e.  Y  |->  [_ A  /  x ]_ Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   [_csb 3390    |-> cmpt 4453
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-opab 4454  df-mpt 4455
This theorem is referenced by:  matgsum  31022
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