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Theorem opeq2 3524
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 2082 . . . . . 6 (A = B → (A V ↔ B V))
21anbi2d 440 . . . . 5 (A = B → ((𝐶 V A V) ↔ (𝐶 V B V)))
3 eqidd 2023 . . . . . . 7 (A = B → {𝐶} = {𝐶})
4 preq2 3422 . . . . . . 7 (A = B → {𝐶, A} = {𝐶, B})
53, 4preq12d 3429 . . . . . 6 (A = B → {{𝐶}, {𝐶, A}} = {{𝐶}, {𝐶, B}})
65eleq2d 2089 . . . . 5 (A = B → (x {{𝐶}, {𝐶, A}} ↔ x {{𝐶}, {𝐶, B}}))
72, 6anbi12d 445 . . . 4 (A = B → (((𝐶 V A V) x {{𝐶}, {𝐶, A}}) ↔ ((𝐶 V B V) x {{𝐶}, {𝐶, B}})))
8 df-3an 875 . . . 4 ((𝐶 V A V x {{𝐶}, {𝐶, A}}) ↔ ((𝐶 V A V) x {{𝐶}, {𝐶, A}}))
9 df-3an 875 . . . 4 ((𝐶 V B V x {{𝐶}, {𝐶, B}}) ↔ ((𝐶 V B V) x {{𝐶}, {𝐶, B}}))
107, 8, 93bitr4g 212 . . 3 (A = B → ((𝐶 V A V x {{𝐶}, {𝐶, A}}) ↔ (𝐶 V B V x {{𝐶}, {𝐶, B}})))
1110abbidv 2137 . 2 (A = B → {x ∣ (𝐶 V A V x {{𝐶}, {𝐶, A}})} = {x ∣ (𝐶 V B V x {{𝐶}, {𝐶, B}})})
12 df-op 3359 . 2 𝐶, A⟩ = {x ∣ (𝐶 V A V x {{𝐶}, {𝐶, A}})}
13 df-op 3359 . 2 𝐶, B⟩ = {x ∣ (𝐶 V B V x {{𝐶}, {𝐶, B}})}
1411, 12, 133eqtr4g 2079 1 (A = B → ⟨𝐶, A⟩ = ⟨𝐶, B⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 873   = wceq 1228   wcel 1374  {cab 2008  Vcvv 2535  {csn 3350  {cpr 3351  cop 3353
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-3an 875  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  df-sn 3356  df-pr 3357  df-op 3359
This theorem is referenced by:  opeq12  3525  opeq2i  3527  opeq2d  3530  oteq2  3533  oteq3  3534  breq2  3742  cbvopab2  3805  cbvopab2v  3808  opthg  3949  eqvinop  3954  opelopabsb  3971  opelxp  4301  opabid2  4394  elrn2g  4452  opeldm  4465  opeldmg  4467  elrn2  4503  opelresg  4546  iss  4581  elimasng  4620  issref  4634  dmsnopg  4719  cnvsng  4733  elxp4  4735  elxp5  4736  funopg  4860  f1osng  5092  tz6.12f  5127  fsn  5260  fsng  5261  fvsng  5284  oveq2  5444  cbvoprab2  5500  ovg  5562  opabex3d  5671  opabex3  5672  op1stg  5700  op2ndg  5701  op1steq  5728  dfoprab4f  5742  tfrlemibxssdm  5862  elreal  6541  ax1rid  6571
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