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Theorem bj-2upleq 32352
Description: Substitution property for (|  - ,  - |). (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-2upleq  |-  ( A  =  B  ->  ( C  =  D  -> (| A,  C|)  = (| B,  D|) ) )

Proof of Theorem bj-2upleq
StepHypRef Expression
1 bj-1upleq 32339 . . 3  |-  ( A  =  B  -> (| A|)  = (| B|) )
2 bj-xtageq 32328 . . 3  |-  ( C  =  D  ->  ( { 1o }  X. tag  C
)  =  ( { 1o }  X. tag  D
) )
3 uneq12 3500 . . . 4  |-  ( ((| A|)  = (| B|)  /\  ( { 1o }  X. tag  C )  =  ( { 1o }  X. tag  D
) )  ->  ((| A|)  u.  ( { 1o }  X. tag  C ) )  =  ((| B|)  u.  ( { 1o }  X. tag  D ) ) )
43ex 434 . . 3  |-  ((| A|)  = (| B|)  ->  (
( { 1o }  X. tag  C )  =  ( { 1o }  X. tag  D )  ->  ((| A|)  u.  ( { 1o }  X. tag  C ) )  =  ((| B|)  u.  ( { 1o }  X. tag  D
) ) ) )
51, 2, 4syl2im 38 . 2  |-  ( A  =  B  ->  ( C  =  D  ->  ((| A|)  u.  ( { 1o }  X. tag  C
) )  =  ((| B|)  u.  ( { 1o }  X. tag  D
) ) ) )
6 df-bj-2upl 32351 . . 3  |- (| A,  C|)  =  ((| A|)  u.  ( { 1o }  X. tag  C
) )
7 df-bj-2upl 32351 . . 3  |- (| B,  D|)  =  ((| B|)  u.  ( { 1o }  X. tag  D
) )
86, 7eqeq12i 2451 . 2  |-  ((| A,  C|)  = (| B,  D|)  <->  ((| A|)  u.  ( { 1o }  X. tag  C
) )  =  ((| B|)  u.  ( { 1o }  X. tag  D
) ) )
95, 8syl6ibr 227 1  |-  ( A  =  B  ->  ( C  =  D  -> (| A,  C|)  = (| B,  D|) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    u. cun 3321   {csn 3872    X. cxp 4833   1oc1o 6905  tag bj-ctag 32314  (|bj-c1upl 32337  (|bj-c2uple 32350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2716  df-v 2969  df-un 3328  df-opab 4346  df-xp 4841  df-bj-sngl 32306  df-bj-tag 32315  df-bj-1upl 32338  df-bj-2upl 32351
This theorem is referenced by:  bj-2uplth  32361
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