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Theorem csbopeq1a 7743
Description: Equality theorem for substitution of a class 𝐴 for an ordered pair 𝑥, 𝑦 in 𝐵 (analogue of csbeq1a 3897). (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 3498 . . . . 5 𝑥 ∈ V
2 vex 3498 . . . . 5 𝑦 ∈ V
31, 2op2ndd 7694 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = 𝑦)
43eqcomd 2827 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝐴))
5 csbeq1a 3897 . . 3 (𝑦 = (2nd𝐴) → 𝐵 = (2nd𝐴) / 𝑦𝐵)
64, 5syl 17 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐵 = (2nd𝐴) / 𝑦𝐵)
71, 2op1std 7693 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝑥)
87eqcomd 2827 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st𝐴))
9 csbeq1a 3897 . . 3 (𝑥 = (1st𝐴) → (2nd𝐴) / 𝑦𝐵 = (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵)
108, 9syl 17 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) / 𝑦𝐵 = (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵)
116, 10eqtr2d 2857 1 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  csb 3883  cop 4567  cfv 6350  1st c1st 7681  2nd c2nd 7682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fv 6358  df-1st 7683  df-2nd 7684
This theorem is referenced by:  dfmpo  7791  f1od2  30451  wdom2d2  39625
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