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Theorem bj-2upleq 32668
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 32655 . . 3 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
2 bj-xtageq 32644 . . 3 (𝐶 = 𝐷 → ({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷))
3 uneq12 3742 . . . 4 ((⦅𝐴⦆ = ⦅𝐵⦆ ∧ ({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷)) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷)))
43ex 450 . . 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 32667 . . 3 𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶))
7 df-bj-2upl 32667 . . 3 𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷))
86, 7eqeq12i 2635 . 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 1480  cun 3554  {csn 4150   × cxp 5074  1𝑜c1o 7501  tag bj-ctag 32630  bj-c1upl 32653  bj-c2uple 32666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-v 3188  df-un 3561  df-opab 4676  df-xp 5082  df-bj-sngl 32622  df-bj-tag 32631  df-bj-1upl 32654  df-bj-2upl 32667
This theorem is referenced by:  bj-2uplth  32677
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