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Theorem eqop 5726
Description: Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
eqop (A (𝑉 × 𝑊) → (A = ⟨B, 𝐶⟩ ↔ ((1stA) = B (2ndA) = 𝐶)))

Proof of Theorem eqop
StepHypRef Expression
1 1st2nd2 5724 . . 3 (A (𝑉 × 𝑊) → A = ⟨(1stA), (2ndA)⟩)
21eqeq1d 2030 . 2 (A (𝑉 × 𝑊) → (A = ⟨B, 𝐶⟩ ↔ ⟨(1stA), (2ndA)⟩ = ⟨B, 𝐶⟩))
3 1stexg 5717 . . 3 (A (𝑉 × 𝑊) → (1stA) V)
4 2ndexg 5718 . . 3 (A (𝑉 × 𝑊) → (2ndA) V)
5 opthg 3949 . . 3 (((1stA) V (2ndA) V) → (⟨(1stA), (2ndA)⟩ = ⟨B, 𝐶⟩ ↔ ((1stA) = B (2ndA) = 𝐶)))
63, 4, 5syl2anc 393 . 2 (A (𝑉 × 𝑊) → (⟨(1stA), (2ndA)⟩ = ⟨B, 𝐶⟩ ↔ ((1stA) = B (2ndA) = 𝐶)))
72, 6bitrd 177 1 (A (𝑉 × 𝑊) → (A = ⟨B, 𝐶⟩ ↔ ((1stA) = B (2ndA) = 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  cop 3353   × cxp 4270  cfv 4829  1st c1st 5688  2nd c2nd 5689
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-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690  df-2nd 5691
This theorem is referenced by:  eqop2  5727  op1steq  5728
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