Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-1upleq Structured version   Visualization version   GIF version

Theorem bj-1upleq 33937
 Description: Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-1upleq (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)

Proof of Theorem bj-1upleq
StepHypRef Expression
1 bj-xtageq 33926 . 2 (𝐴 = 𝐵 → ({∅} × tag 𝐴) = ({∅} × tag 𝐵))
2 df-bj-1upl 33936 . 2 𝐴⦆ = ({∅} × tag 𝐴)
3 df-bj-1upl 33936 . 2 𝐵⦆ = ({∅} × tag 𝐵)
41, 2, 33eqtr4g 2858 1 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1525  ∅c0 4217  {csn 4478   × cxp 5448  tag bj-ctag 33912  ⦅bj-c1upl 33935 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rex 3113  df-v 3442  df-un 3870  df-opab 5031  df-xp 5456  df-bj-sngl 33904  df-bj-tag 33913  df-bj-1upl 33936 This theorem is referenced by:  bj-1uplth  33945  bj-2upleq  33950
 Copyright terms: Public domain W3C validator