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Theorem opthg 3945
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 3519 . . . 4 (x = A → ⟨x, y⟩ = ⟨A, y⟩)
21eqeq1d 2026 . . 3 (x = A → (⟨x, y⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨A, y⟩ = ⟨𝐶, 𝐷⟩))
3 eqeq1 2024 . . . 4 (x = A → (x = 𝐶A = 𝐶))
43anbi1d 441 . . 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 3520 . . . 4 (y = B → ⟨A, y⟩ = ⟨A, B⟩)
76eqeq1d 2026 . . 3 (y = B → (⟨A, y⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨A, B⟩ = ⟨𝐶, 𝐷⟩))
8 eqeq1 2024 . . . 4 (y = B → (y = 𝐷B = 𝐷))
98anbi2d 440 . . 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 2534 . . 3 x V
12 vex 2534 . . 3 y V
1311, 12opth 3944 . 2 (⟨x, y⟩ = ⟨𝐶, 𝐷⟩ ↔ (x = 𝐶 y = 𝐷))
145, 10, 13vtocl2g 2590 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 1226   wcel 1370  cop 3349
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355
This theorem is referenced by:  opthg2  3946  xpopth  5721  eqop  5722  preqlu  6320
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