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

Theorem bj-tageq 33295
 Description: Substitution property for tag. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-tageq (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)

Proof of Theorem bj-tageq
StepHypRef Expression
1 bj-sngleq 33286 . . 3 (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵)
21uneq1d 3917 . 2 (𝐴 = 𝐵 → (sngl 𝐴 ∪ {∅}) = (sngl 𝐵 ∪ {∅}))
3 df-bj-tag 33294 . 2 tag 𝐴 = (sngl 𝐴 ∪ {∅})
4 df-bj-tag 33294 . 2 tag 𝐵 = (sngl 𝐵 ∪ {∅})
52, 3, 43eqtr4g 2830 1 (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631   ∪ cun 3721  ∅c0 4063  {csn 4317  sngl bj-csngl 33284  tag bj-ctag 33293 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-v 3353  df-un 3728  df-bj-sngl 33285  df-bj-tag 33294 This theorem is referenced by:  bj-xtageq  33307
 Copyright terms: Public domain W3C validator