ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opeq2 GIF version

Theorem opeq2 3577
Description: Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opeq2 (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Proof of Theorem opeq2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2116 . . . . . 6 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
21anbi2d 445 . . . . 5 (𝐴 = 𝐵 → ((𝐶 ∈ V ∧ 𝐴 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐵 ∈ V)))
3 eqidd 2057 . . . . . . 7 (𝐴 = 𝐵 → {𝐶} = {𝐶})
4 preq2 3475 . . . . . . 7 (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
53, 4preq12d 3482 . . . . . 6 (𝐴 = 𝐵 → {{𝐶}, {𝐶, 𝐴}} = {{𝐶}, {𝐶, 𝐵}})
65eleq2d 2123 . . . . 5 (𝐴 = 𝐵 → (𝑥 ∈ {{𝐶}, {𝐶, 𝐴}} ↔ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}}))
72, 6anbi12d 450 . . . 4 (𝐴 = 𝐵 → (((𝐶 ∈ V ∧ 𝐴 ∈ V) ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}}) ↔ ((𝐶 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}})))
8 df-3an 898 . . . 4 ((𝐶 ∈ V ∧ 𝐴 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}}) ↔ ((𝐶 ∈ V ∧ 𝐴 ∈ V) ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}}))
9 df-3an 898 . . . 4 ((𝐶 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}}) ↔ ((𝐶 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}}))
107, 8, 93bitr4g 216 . . 3 (𝐴 = 𝐵 → ((𝐶 ∈ V ∧ 𝐴 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}}) ↔ (𝐶 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}})))
1110abbidv 2171 . 2 (𝐴 = 𝐵 → {𝑥 ∣ (𝐶 ∈ V ∧ 𝐴 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}})} = {𝑥 ∣ (𝐶 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}})})
12 df-op 3411 . 2 𝐶, 𝐴⟩ = {𝑥 ∣ (𝐶 ∈ V ∧ 𝐴 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}})}
13 df-op 3411 . 2 𝐶, 𝐵⟩ = {𝑥 ∣ (𝐶 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}})}
1411, 12, 133eqtr4g 2113 1 (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896   = wceq 1259  wcel 1409  {cab 2042  Vcvv 2574  {csn 3402  {cpr 3403  cop 3405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411
This theorem is referenced by:  opeq12  3578  opeq2i  3580  opeq2d  3583  oteq2  3586  oteq3  3587  breq2  3795  cbvopab2  3858  cbvopab2v  3861  opthg  4002  eqvinop  4007  opelopabsb  4024  opelxp  4401  opabid2  4494  elrn2g  4552  opeldm  4565  opeldmg  4567  elrn2  4603  opelresg  4646  iss  4681  elimasng  4720  issref  4734  dmsnopg  4819  cnvsng  4833  elxp4  4835  elxp5  4836  dffun5r  4941  funopg  4961  f1osng  5194  tz6.12f  5229  fsn  5362  fsng  5363  fvsng  5386  oveq2  5547  cbvoprab2  5604  ovg  5666  opabex3d  5775  opabex3  5776  op1stg  5804  op2ndg  5805  op1steq  5832  dfoprab4f  5846  tfrlemibxssdm  5971  xpsnen  6325  xpassen  6334  elreal  6962  ax1rid  7008  fseq1p1m1  9057
  Copyright terms: Public domain W3C validator