Theorem bj-2upleq 33000

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

Step	Hyp	Ref	Expression
1		bj-1upleq 32987	. . 3 |- ( A = B -> (| A|) = (| B|) )
2		bj-xtageq 32976	. . 3 |- ( C = D -> ( { 1o } X. tag C ) = ( { 1o } X. tag D ) )
3		uneq12 3762	. . . 4 |- ( ((| A|) = (| B|) /\ ( { 1o } X. tag C ) = ( { 1o } X. tag D ) ) -> ((| A|) u. ( { 1o } X. tag C ) ) = ((| B|) u. ( { 1o } X. tag D ) ) )
4	3	ex 450	. . 3 |- ((| A|) = (| B|) -> ( ( { 1o } X. tag C ) = ( { 1o } X. tag D ) -> ((| A|) u. ( { 1o } X. tag C ) ) = ((| B|) u. ( { 1o } X. tag D ) ) ) )
5	1, 2, 4	syl2im 40	. 2 |- ( A = B -> ( C = D -> ((| A|) u. ( { 1o } X. tag C ) ) = ((| B|) u. ( { 1o } X. tag D ) ) ) )
6		df-bj-2upl 32999	. . 3 |- (| A, C|) = ((| A|) u. ( { 1o } X. tag C ) )
7		df-bj-2upl 32999	. . 3 |- (| B, D|) = ((| B|) u. ( { 1o } X. tag D ) )
8	6, 7	eqeq12i 2636	. 2 |- ((| A, C|) = (| B, D|) <-> ((| A|) u. ( { 1o } X. tag C ) ) = ((| B|) u. ( { 1o } X. tag D ) ) )
9	5, 8	syl6ibr 242	1 |- ( A = B -> ( C = D -> (| A, C|) = (| B, D|) ) )

Syntax hints:   -> wi 4   = wceq 1483   u. cun 3572   {csn 4177   X. cxp 5112   1oc1o 7553  tag bj-ctag 32962  (|bj-c1upl 32985  (|bj-c2uple 32998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-un 3579  df-opab 4713  df-xp 5120  df-bj-sngl 32954  df-bj-tag 32963  df-bj-1upl 32986  df-bj-2upl 32999

This theorem is referenced by:  bj-2uplth  33009