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Theorem bj-2upleq 33576
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 33563 . . 3 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
2 bj-xtageq 33552 . . 3 (𝐶 = 𝐷 → ({1o} × tag 𝐶) = ({1o} × tag 𝐷))
3 uneq12 3985 . . . 4 ((⦅𝐴⦆ = ⦅𝐵⦆ ∧ ({1o} × tag 𝐶) = ({1o} × tag 𝐷)) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))
43ex 403 . . 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 33575 . . 3 𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐶))
7 df-bj-2upl 33575 . . 3 𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷))
86, 7eqeq12i 2792 . 2 (⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆ ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))
95, 8syl6ibr 244 1 (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  cun 3790  {csn 4398   × cxp 5355  1oc1o 7838  tag bj-ctag 33538  bj-c1upl 33561  bj-c2uple 33574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rex 3096  df-v 3400  df-un 3797  df-opab 4951  df-xp 5363  df-bj-sngl 33530  df-bj-tag 33539  df-bj-1upl 33562  df-bj-2upl 33575
This theorem is referenced by:  bj-2uplth  33585
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