Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-2upleq Structured version   Visualization version   GIF version

Theorem bj-2upleq 33125
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 33112 . . 3 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
2 bj-xtageq 33101 . . 3 (𝐶 = 𝐷 → ({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷))
3 uneq12 3795 . . . 4 ((⦅𝐴⦆ = ⦅𝐵⦆ ∧ ({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷)) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷)))
43ex 449 . . 3 (⦅𝐴⦆ = ⦅𝐵⦆ → (({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷))))
51, 2, 4syl2im 40 . 2 (𝐴 = 𝐵 → (𝐶 = 𝐷 → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷))))
6 df-bj-2upl 33124 . . 3 𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶))
7 df-bj-2upl 33124 . . 3 𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷))
86, 7eqeq12i 2665 . 2 (⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆ ↔ (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷)))
95, 8syl6ibr 242 1 (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  cun 3605  {csn 4210   × cxp 5141  1𝑜c1o 7598  tag bj-ctag 33087  bj-c1upl 33110  bj-c2uple 33123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-un 3612  df-opab 4746  df-xp 5149  df-bj-sngl 33079  df-bj-tag 33088  df-bj-1upl 33111  df-bj-2upl 33124
This theorem is referenced by:  bj-2uplth  33134
  Copyright terms: Public domain W3C validator