Theorem bj-2upleq 37260

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 37247	. . 3 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
2		bj-xtageq 37236	. . 3 ⊢ (𝐶 = 𝐷 → ({1o} × tag 𝐶) = ({1o} × tag 𝐷))
3		uneq12 4117	. . . 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 37259	. . 3 ⊢ ⦅𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐶))
7		df-bj-2upl 37259	. . 3 ⊢ ⦅𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷))
8	6, 7	eqeq12i 2755	. 2 ⊢ (⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆ ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))
9	5, 8	imbitrrdi 252	1 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))

Syntax hints:   → wi 4   = wceq 1542   ∪ cun 3901  {csn 4582   × cxp 5630  1_oc1o 8400  tag bj-ctag 37222  ⦅bj-c1upl 37245  ⦅bj-c2uple 37258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3063  df-v 3444  df-un 3908  df-opab 5163  df-xp 5638  df-bj-sngl 37214  df-bj-tag 37223  df-bj-1upl 37246  df-bj-2upl 37259

This theorem is referenced by:  bj-2uplth  37269