 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbopeq1a GIF version

Theorem csbopeq1a 5841
 Description: Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analog of csbeq1a 2887). (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 2577 . . . . 5 𝑥 ∈ V
2 vex 2577 . . . . 5 𝑦 ∈ V
31, 2op2ndd 5803 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = 𝑦)
43eqcomd 2061 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝐴))
5 csbeq1a 2887 . . 3 (𝑦 = (2nd𝐴) → 𝐵 = (2nd𝐴) / 𝑦𝐵)
64, 5syl 14 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐵 = (2nd𝐴) / 𝑦𝐵)
71, 2op1std 5802 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝑥)
87eqcomd 2061 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st𝐴))
9 csbeq1a 2887 . . 3 (𝑥 = (1st𝐴) → (2nd𝐴) / 𝑦𝐵 = (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵)
108, 9syl 14 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) / 𝑦𝐵 = (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵)
116, 10eqtr2d 2089 1 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) / 𝑥(2nd𝐴) / 𝑦𝐵 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259  ⦋csb 2879  ⟨cop 3405  ‘cfv 4929  1st c1st 5792  2nd c2nd 5793 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fv 4937  df-1st 5794  df-2nd 5795 This theorem is referenced by:  dfmpt2  5871  f1od2  5883
 Copyright terms: Public domain W3C validator