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Theorem opthg 3966
Description: Ordered pair theorem. 𝐶 and 𝐷 are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opthg ((A 𝑉 B 𝑊) → (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 B = 𝐷)))

Proof of Theorem opthg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3540 . . . 4 (x = A → ⟨x, y⟩ = ⟨A, y⟩)
21eqeq1d 2045 . . 3 (x = A → (⟨x, y⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨A, y⟩ = ⟨𝐶, 𝐷⟩))
3 eqeq1 2043 . . . 4 (x = A → (x = 𝐶A = 𝐶))
43anbi1d 438 . . 3 (x = A → ((x = 𝐶 y = 𝐷) ↔ (A = 𝐶 y = 𝐷)))
52, 4bibi12d 224 . 2 (x = A → ((⟨x, y⟩ = ⟨𝐶, 𝐷⟩ ↔ (x = 𝐶 y = 𝐷)) ↔ (⟨A, y⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 y = 𝐷))))
6 opeq2 3541 . . . 4 (y = B → ⟨A, y⟩ = ⟨A, B⟩)
76eqeq1d 2045 . . 3 (y = B → (⟨A, y⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨A, B⟩ = ⟨𝐶, 𝐷⟩))
8 eqeq1 2043 . . . 4 (y = B → (y = 𝐷B = 𝐷))
98anbi2d 437 . . 3 (y = B → ((A = 𝐶 y = 𝐷) ↔ (A = 𝐶 B = 𝐷)))
107, 9bibi12d 224 . 2 (y = B → ((⟨A, y⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 y = 𝐷)) ↔ (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 B = 𝐷))))
11 vex 2554 . . 3 x V
12 vex 2554 . . 3 y V
1311, 12opth 3965 . 2 (⟨x, y⟩ = ⟨𝐶, 𝐷⟩ ↔ (x = 𝐶 y = 𝐷))
145, 10, 13vtocl2g 2611 1 ((A 𝑉 B 𝑊) → (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 B = 𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  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-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
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-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by:  opthg2  3967  xpopth  5744  eqop  5745  preqlu  6455  cauappcvgprlemladd  6630
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