ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opeq1 Structured version   GIF version

Theorem opeq1 3540
Description: Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opeq1 (A = B → ⟨A, 𝐶⟩ = ⟨B, 𝐶⟩)

Proof of Theorem opeq1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . . . . . 6 (A = B → (A V ↔ B V))
21anbi1d 438 . . . . 5 (A = B → ((A V 𝐶 V) ↔ (B V 𝐶 V)))
3 sneq 3378 . . . . . . 7 (A = B → {A} = {B})
4 preq1 3438 . . . . . . 7 (A = B → {A, 𝐶} = {B, 𝐶})
53, 4preq12d 3446 . . . . . 6 (A = B → {{A}, {A, 𝐶}} = {{B}, {B, 𝐶}})
65eleq2d 2104 . . . . 5 (A = B → (x {{A}, {A, 𝐶}} ↔ x {{B}, {B, 𝐶}}))
72, 6anbi12d 442 . . . 4 (A = B → (((A V 𝐶 V) x {{A}, {A, 𝐶}}) ↔ ((B V 𝐶 V) x {{B}, {B, 𝐶}})))
8 df-3an 886 . . . 4 ((A V 𝐶 V x {{A}, {A, 𝐶}}) ↔ ((A V 𝐶 V) x {{A}, {A, 𝐶}}))
9 df-3an 886 . . . 4 ((B V 𝐶 V x {{B}, {B, 𝐶}}) ↔ ((B V 𝐶 V) x {{B}, {B, 𝐶}}))
107, 8, 93bitr4g 212 . . 3 (A = B → ((A V 𝐶 V x {{A}, {A, 𝐶}}) ↔ (B V 𝐶 V x {{B}, {B, 𝐶}})))
1110abbidv 2152 . 2 (A = B → {x ∣ (A V 𝐶 V x {{A}, {A, 𝐶}})} = {x ∣ (B V 𝐶 V x {{B}, {B, 𝐶}})})
12 df-op 3376 . 2 A, 𝐶⟩ = {x ∣ (A V 𝐶 V x {{A}, {A, 𝐶}})}
13 df-op 3376 . 2 B, 𝐶⟩ = {x ∣ (B V 𝐶 V x {{B}, {B, 𝐶}})}
1411, 12, 133eqtr4g 2094 1 (A = B → ⟨A, 𝐶⟩ = ⟨B, 𝐶⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  {cab 2023  Vcvv 2551  {csn 3367  {cpr 3368  cop 3370
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-3an 886  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  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by:  opeq12  3542  opeq1i  3543  opeq1d  3546  oteq1  3549  breq1  3758  cbvopab1  3821  cbvopab1s  3823  opthg  3966  eqvinop  3971  opelopabsb  3988  opelxp  4317  elvvv  4346  opabid2  4410  opeliunxp2  4419  elsnres  4590  elimasng  4636  rnxpid  4698  dmsnopg  4735  cnvsng  4749  elxp4  4751  elxp5  4752  funopg  4877  f1osng  5110  dmfco  5184  fvelrn  5241  fsng  5279  fvsng  5302  funfvima3  5335  oveq1  5462  oprabid  5480  dfoprab2  5494  cbvoprab1  5518  opabex3d  5690  opabex3  5691  op1stg  5719  op2ndg  5720  dfoprab4f  5761  fundmen  6222  xpsnen  6231  xpassen  6240  ltexnqq  6391  archnqq  6400  prarloclemarch2  6402  prarloclemlo  6476  prarloclem3  6479  prarloclem5  6482  pitonn  6704
  Copyright terms: Public domain W3C validator