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Theorem eqvinop 3950
 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 3947 . . . . . . 7 (⟨x, y⟩ = ⟨B, 𝐶⟩ ↔ (x = B y = 𝐶))
43anbi2i 433 . . . . . 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 383 . . . . . 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 1474 . . . 4 (y(A = ⟨x, yx, y⟩ = ⟨B, 𝐶⟩) ↔ y(x = B (y = 𝐶 A = ⟨x, y⟩)))
9 19.42v 1764 . . . 4 (y(x = B (y = 𝐶 A = ⟨x, y⟩)) ↔ (x = B y(y = 𝐶 A = ⟨x, y⟩)))
10 opeq2 3520 . . . . . . 7 (y = 𝐶 → ⟨x, y⟩ = ⟨x, 𝐶⟩)
1110eqeq2d 2029 . . . . . 6 (y = 𝐶 → (A = ⟨x, y⟩ ↔ A = ⟨x, 𝐶⟩))
122, 11ceqsexv 2566 . . . . 5 (y(y = 𝐶 A = ⟨x, y⟩) ↔ A = ⟨x, 𝐶⟩)
1312anbi2i 433 . . . 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 1474 . 2 (xy(A = ⟨x, yx, y⟩ = ⟨B, 𝐶⟩) ↔ x(x = B A = ⟨x, 𝐶⟩))
16 opeq1 3519 . . . 4 (x = B → ⟨x, 𝐶⟩ = ⟨B, 𝐶⟩)
1716eqeq2d 2029 . . 3 (x = B → (A = ⟨x, 𝐶⟩ ↔ A = ⟨B, 𝐶⟩))
181, 17ceqsexv 2566 . 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 1226  ∃wex 1358   ∈ wcel 1370  Vcvv 2531  ⟨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:  copsexg  3951  ralxpf  4405  rexxpf  4406  oprabid  5457
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