Theorem uneq1 3412

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 2414	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
2	1	orbi1d 683	. . 3 ⊢ (A = B → ((x ∈ A ∨ x ∈ C) ↔ (x ∈ B ∨ x ∈ C)))
3		elun 3221	. . 3 ⊢ (x ∈ (A ∪ C) ↔ (x ∈ A ∨ x ∈ C))
4		elun 3221	. . 3 ⊢ (x ∈ (B ∪ C) ↔ (x ∈ B ∨ x ∈ C))
5	2, 3, 4	3bitr4g 279	. 2 ⊢ (A = B → (x ∈ (A ∪ C) ↔ x ∈ (B ∪ C)))
6	5	eqrdv 2351	1 ⊢ (A = B → (A ∪ C) = (B ∪ C))

Syntax hints:   → wi 4   ∨ wo 357   = wceq 1642   ∈ wcel 1710   ∪ cun 3208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215

This theorem is referenced by:  uneq2  3413  uneq12  3414  uneq1i  3415  uneq1d  3418  unineq  3506  adj11  3890  uniprg  3907  pwadjoin  4120  eladdci  4400  elsuci  4415  addcass  4416  nnsucelr  4429  nnadjoin  4521  phi011  4600  cupvalg  5813  el2c  6192  nmembers1lem3  6271