Theorem bj-2upleq 37365

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

Step	Hyp	Ref	Expression
1		bj-1upleq 37352	. . 3 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
2		bj-xtageq 37341	. . 3 ⊢ (𝐶 = 𝐷 → ({1o} × tag 𝐶) = ({1o} × tag 𝐷))
3		uneq12 4093	. . . 4 ⊢ ((⦅𝐴⦆ = ⦅𝐵⦆ ∧ ({1o} × tag 𝐶) = ({1o} × tag 𝐷)) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))
4	3	ex 413	. . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → (({1o} × tag 𝐶) = ({1o} × tag 𝐷) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷))))
5	1, 2, 4	syl2im 40	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷))))
6		df-bj-2upl 37364	. . 3 ⊢ ⦅𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐶))
7		df-bj-2upl 37364	. . 3 ⊢ ⦅𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷))
8	6, 7	eqeq12i 2757	. 2 ⊢ (⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆ ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))
9	5, 8	imbitrrdi 253	1 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))

Syntax hints:   → wi 4   = wceq 1547   ∪ cun 3881  {csn 4555   × cxp 5616  1_oc1o 8388  tag bj-ctag 37327  ⦅bj-c1upl 37350  ⦅bj-c2uple 37363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rex 3064  df-v 3433  df-un 3888  df-opab 5135  df-xp 5624  df-bj-sngl 37319  df-bj-tag 37328  df-bj-1upl 37351  df-bj-2upl 37364

This theorem is referenced by:  bj-2uplth  37374