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Theorem csbmpt2 4782
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 4781 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( y  e.  Y  |->  Z )  =  ( y  e. 
[_ A  /  x ]_ Y  |->  [_ A  /  x ]_ Z ) )
2 csbconstg 3448 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ Y  =  Y )
32mpteq1d 4528 . 2  |-  ( A  e.  V  ->  (
y  e.  [_ A  /  x ]_ Y  |->  [_ A  /  x ]_ Z
)  =  ( y  e.  Y  |->  [_ A  /  x ]_ Z ) )
41, 3eqtrd 2508 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 1379    e. wcel 1767   [_csb 3435    |-> cmpt 4505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-mpt 4507
This theorem is referenced by:  matgsum  18734
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