ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uneq1 GIF version

Theorem uneq1 3282
Description: Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
uneq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem uneq1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2241 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21orbi1d 791 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
3 elun 3276 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elun 3276 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
52, 3, 43bitr4g 223 . 2 (𝐴 = 𝐵 → (𝑥 ∈ (𝐴𝐶) ↔ 𝑥 ∈ (𝐵𝐶)))
65eqrdv 2175 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 708   = wceq 1353  wcel 2148  cun 3127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133
This theorem is referenced by:  uneq2  3283  uneq12  3284  uneq1i  3285  uneq1d  3288  prprc1  3699  uniprg  3822  unexb  4438  relresfld  5153  relcoi1  5155  rdgeq2  6366  xpider  6599  findcard2  6882  findcard2s  6883  unfiexmid  6910  bdunexb  14294  bj-unexg  14295  exmid1stab  14372
  Copyright terms: Public domain W3C validator