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Theorem bj-2upleq 33871
 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 33858 . . 3 (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
2 bj-xtageq 33847 . . 3 (𝐶 = 𝐷 → ({1o} × tag 𝐶) = ({1o} × tag 𝐷))
3 uneq12 4017 . . . 4 ((⦅𝐴⦆ = ⦅𝐵⦆ ∧ ({1o} × tag 𝐶) = ({1o} × tag 𝐷)) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))
43ex 405 . . 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 33870 . . 3 𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐶))
7 df-bj-2upl 33870 . . 3 𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷))
86, 7eqeq12i 2786 . 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 1507   ∪ cun 3821  {csn 4435   × cxp 5401  1oc1o 7896  tag bj-ctag 33833  ⦅bj-c1upl 33856  ⦅bj-c2uple 33869 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rex 3088  df-v 3411  df-un 3828  df-opab 4988  df-xp 5409  df-bj-sngl 33825  df-bj-tag 33834  df-bj-1upl 33857  df-bj-2upl 33870 This theorem is referenced by:  bj-2uplth  33880
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