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Theorem eqop 5745
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 5743 . . 3 (A (𝑉 × 𝑊) → A = ⟨(1stA), (2ndA)⟩)
21eqeq1d 2045 . 2 (A (𝑉 × 𝑊) → (A = ⟨B, 𝐶⟩ ↔ ⟨(1stA), (2ndA)⟩ = ⟨B, 𝐶⟩))
3 1stexg 5736 . . 3 (A (𝑉 × 𝑊) → (1stA) V)
4 2ndexg 5737 . . 3 (A (𝑉 × 𝑊) → (2ndA) V)
5 opthg 3966 . . 3 (((1stA) V (2ndA) V) → (⟨(1stA), (2ndA)⟩ = ⟨B, 𝐶⟩ ↔ ((1stA) = B (2ndA) = 𝐶)))
63, 4, 5syl2anc 391 . 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 1242   wcel 1390  Vcvv 2551  cop 3370   × cxp 4286  cfv 4845  1st c1st 5707  2nd c2nd 5708
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-13 1401  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  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by:  eqop2  5746  op1steq  5747
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