Theorem bj-2upleq 33125

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 33112	. . 3 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
2		bj-xtageq 33101	. . 3 ⊢ (𝐶 = 𝐷 → ({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷))
3		uneq12 3795	. . . 4 ⊢ ((⦅𝐴⦆ = ⦅𝐵⦆ ∧ ({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷)) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷)))
4	3	ex 449	. . 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 33124	. . 3 ⊢ ⦅𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶))
7		df-bj-2upl 33124	. . 3 ⊢ ⦅𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷))
8	6, 7	eqeq12i 2665	. 2 ⊢ (⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆ ↔ (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷)))
9	5, 8	syl6ibr 242	1 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))

Syntax hints:   → wi 4   = wceq 1523   ∪ cun 3605  {csn 4210   × cxp 5141  1_𝑜c1o 7598  tag bj-ctag 33087  ⦅bj-c1upl 33110  ⦅bj-c2uple 33123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-un 3612  df-opab 4746  df-xp 5149  df-bj-sngl 33079  df-bj-tag 33088  df-bj-1upl 33111  df-bj-2upl 33124

This theorem is referenced by:  bj-2uplth  33134