Theorem uneq1 3067

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 2083	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
2	1	orbi1d 692	. . 3 ⊢ (A = B → ((x ∈ A ∨ x ∈ 𝐶) ↔ (x ∈ B ∨ x ∈ 𝐶)))
3		elun 3061	. . 3 ⊢ (x ∈ (A ∪ 𝐶) ↔ (x ∈ A ∨ x ∈ 𝐶))
4		elun 3061	. . 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 2020	1 ⊢ (A = B → (A ∪ 𝐶) = (B ∪ 𝐶))

Syntax hints:   → wi 4   ∨ wo 616   = wceq 1228   ∈ wcel 1374   ∪ cun 2892

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899

This theorem is referenced by:  uneq2  3068  uneq12  3069  uneq1i  3070  uneq1d  3073  prprc1  3452  uniprg  3569  unexb  4127  relresfld  4774  relcoi1  4776  rdgeq2  5880  xpiderm  6088  bdunexb  7143  bj-unexg  7144