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Theorem csbopeq1a 6054
Description: Equality theorem for substitution of a class 𝐴 for an ordered pair 𝑥, 𝑦 in 𝐵 (analog of csbeq1a 2983). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
csbopeq1a (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵 = 𝐵)

Proof of Theorem csbopeq1a
StepHypRef Expression
1 vex 2663 . . . . 5 𝑥 ∈ V
2 vex 2663 . . . . 5 𝑦 ∈ V
31, 2op2ndd 6015 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = 𝑦)
43eqcomd 2123 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝐴))
5 csbeq1a 2983 . . 3 (𝑦 = (2nd𝐴) → 𝐵 = (2nd𝐴) / 𝑦𝐵)
64, 5syl 14 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐵 = (2nd𝐴) / 𝑦𝐵)
71, 2op1std 6014 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝑥)
87eqcomd 2123 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st𝐴))
9 csbeq1a 2983 . . 3 (𝑥 = (1st𝐴) → (2nd𝐴) / 𝑦𝐵 = (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵)
108, 9syl 14 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) / 𝑦𝐵 = (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵)
116, 10eqtr2d 2151 1 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  csb 2975  cop 3500  cfv 5093  1st c1st 6004  2nd c2nd 6005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fv 5101  df-1st 6006  df-2nd 6007
This theorem is referenced by:  dfmpo  6088  f1od2  6100
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