Theorem bj-2upleq 36383

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 36370	. . 3 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
2		bj-xtageq 36359	. . 3 ⊢ (𝐶 = 𝐷 → ({1o} × tag 𝐶) = ({1o} × tag 𝐷))
3		uneq12 4150	. . . 4 ⊢ ((⦅𝐴⦆ = ⦅𝐵⦆ ∧ ({1o} × tag 𝐶) = ({1o} × tag 𝐷)) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))
4	3	ex 412	. . 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 36382	. . 3 ⊢ ⦅𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐶))
7		df-bj-2upl 36382	. . 3 ⊢ ⦅𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷))
8	6, 7	eqeq12i 2742	. 2 ⊢ (⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆ ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))
9	5, 8	imbitrrdi 251	1 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))

Syntax hints:   → wi 4   = wceq 1533   ∪ cun 3938  {csn 4620   × cxp 5664  1_oc1o 8454  tag bj-ctag 36345  ⦅bj-c1upl 36368  ⦅bj-c2uple 36381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rex 3063  df-v 3468  df-un 3945  df-opab 5201  df-xp 5672  df-bj-sngl 36337  df-bj-tag 36346  df-bj-1upl 36369  df-bj-2upl 36382

This theorem is referenced by:  bj-2uplth  36392