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Theorem eqvinop 3971
Description: A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
Hypotheses
Ref Expression
eqvinop.1 B V
eqvinop.2 𝐶 V
Assertion
Ref Expression
eqvinop (A = ⟨B, 𝐶⟩ ↔ xy(A = ⟨x, yx, y⟩ = ⟨B, 𝐶⟩))
Distinct variable groups:   x,y,A   x,B,y   x,𝐶,y

Proof of Theorem eqvinop
StepHypRef Expression
1 eqvinop.1 . . . . . . . 8 B V
2 eqvinop.2 . . . . . . . 8 𝐶 V
31, 2opth2 3968 . . . . . . 7 (⟨x, y⟩ = ⟨B, 𝐶⟩ ↔ (x = B y = 𝐶))
43anbi2i 430 . . . . . 6 ((A = ⟨x, yx, y⟩ = ⟨B, 𝐶⟩) ↔ (A = ⟨x, y (x = B y = 𝐶)))
5 ancom 253 . . . . . 6 ((A = ⟨x, y (x = B y = 𝐶)) ↔ ((x = B y = 𝐶) A = ⟨x, y⟩))
6 anass 381 . . . . . 6 (((x = B y = 𝐶) A = ⟨x, y⟩) ↔ (x = B (y = 𝐶 A = ⟨x, y⟩)))
74, 5, 63bitri 195 . . . . 5 ((A = ⟨x, yx, y⟩ = ⟨B, 𝐶⟩) ↔ (x = B (y = 𝐶 A = ⟨x, y⟩)))
87exbii 1493 . . . 4 (y(A = ⟨x, yx, y⟩ = ⟨B, 𝐶⟩) ↔ y(x = B (y = 𝐶 A = ⟨x, y⟩)))
9 19.42v 1783 . . . 4 (y(x = B (y = 𝐶 A = ⟨x, y⟩)) ↔ (x = B y(y = 𝐶 A = ⟨x, y⟩)))
10 opeq2 3541 . . . . . . 7 (y = 𝐶 → ⟨x, y⟩ = ⟨x, 𝐶⟩)
1110eqeq2d 2048 . . . . . 6 (y = 𝐶 → (A = ⟨x, y⟩ ↔ A = ⟨x, 𝐶⟩))
122, 11ceqsexv 2587 . . . . 5 (y(y = 𝐶 A = ⟨x, y⟩) ↔ A = ⟨x, 𝐶⟩)
1312anbi2i 430 . . . 4 ((x = B y(y = 𝐶 A = ⟨x, y⟩)) ↔ (x = B A = ⟨x, 𝐶⟩))
148, 9, 133bitri 195 . . 3 (y(A = ⟨x, yx, y⟩ = ⟨B, 𝐶⟩) ↔ (x = B A = ⟨x, 𝐶⟩))
1514exbii 1493 . 2 (xy(A = ⟨x, yx, y⟩ = ⟨B, 𝐶⟩) ↔ x(x = B A = ⟨x, 𝐶⟩))
16 opeq1 3540 . . . 4 (x = B → ⟨x, 𝐶⟩ = ⟨B, 𝐶⟩)
1716eqeq2d 2048 . . 3 (x = B → (A = ⟨x, 𝐶⟩ ↔ A = ⟨B, 𝐶⟩))
181, 17ceqsexv 2587 . 2 (x(x = B A = ⟨x, 𝐶⟩) ↔ A = ⟨B, 𝐶⟩)
1915, 18bitr2i 174 1 (A = ⟨B, 𝐶⟩ ↔ xy(A = ⟨x, yx, y⟩ = ⟨B, 𝐶⟩))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  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-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:  copsexg  3972  ralxpf  4425  rexxpf  4426  oprabid  5480
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