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Theorem bj-2upleq 34765
 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 34752 . . 3 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
2 bj-xtageq 34741 . . 3 (𝐶 = 𝐷 → ({1o} × tag 𝐶) = ({1o} × tag 𝐷))
3 uneq12 4066 . . . 4 ((⦅𝐴⦆ = ⦅𝐵⦆ ∧ ({1o} × tag 𝐶) = ({1o} × tag 𝐷)) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))
43ex 416 . . 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 34764 . . 3 𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐶))
7 df-bj-2upl 34764 . . 3 𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷))
86, 7eqeq12i 2774 . 2 (⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆ ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))
95, 8syl6ibr 255 1 (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∪ cun 3859  {csn 4526   × cxp 5527  1oc1o 8112  tag bj-ctag 34727  ⦅bj-c1upl 34750  ⦅bj-c2uple 34763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-rex 3077  df-v 3412  df-un 3866  df-opab 5100  df-xp 5535  df-bj-sngl 34719  df-bj-tag 34728  df-bj-1upl 34751  df-bj-2upl 34764 This theorem is referenced by:  bj-2uplth  34774
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