Theorem uneq1 3084

Description: Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
uneq1	⊢ (A = B → (A ∪ 𝐶) = (B ∪ 𝐶))

Proof of Theorem uneq1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2098	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
2	1	orbi1d 704	. . 3 ⊢ (A = B → ((x ∈ A ∨ x ∈ 𝐶) ↔ (x ∈ B ∨ x ∈ 𝐶)))
3		elun 3078	. . 3 ⊢ (x ∈ (A ∪ 𝐶) ↔ (x ∈ A ∨ x ∈ 𝐶))
4		elun 3078	. . 3 ⊢ (x ∈ (B ∪ 𝐶) ↔ (x ∈ B ∨ x ∈ 𝐶))
5	2, 3, 4	3bitr4g 212	. 2 ⊢ (A = B → (x ∈ (A ∪ 𝐶) ↔ x ∈ (B ∪ 𝐶)))
6	5	eqrdv 2035	1 ⊢ (A = B → (A ∪ 𝐶) = (B ∪ 𝐶))

Syntax hints:   → wi 4   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ∪ cun 2909

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916

This theorem is referenced by:  uneq2  3085  uneq12  3086  uneq1i  3087  uneq1d  3090  prprc1  3469  uniprg  3586  unexb  4143  relresfld  4790  relcoi1  4792  rdgeq2  5899  xpiderm  6113  bdunexb  9351  bj-unexg  9352