ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcopeq1a GIF version

Theorem sbcopeq1a 6163
Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2964 that avoids the existential quantifiers of copsexg 4227). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
sbcopeq1a (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑𝜑))

Proof of Theorem sbcopeq1a
StepHypRef Expression
1 vex 2733 . . . . 5 𝑥 ∈ V
2 vex 2733 . . . . 5 𝑦 ∈ V
31, 2op2ndd 6125 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = 𝑦)
43eqcomd 2176 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd𝐴))
5 sbceq1a 2964 . . 3 (𝑦 = (2nd𝐴) → (𝜑[(2nd𝐴) / 𝑦]𝜑))
64, 5syl 14 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑[(2nd𝐴) / 𝑦]𝜑))
71, 2op1std 6124 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝑥)
87eqcomd 2176 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st𝐴))
9 sbceq1a 2964 . . 3 (𝑥 = (1st𝐴) → ([(2nd𝐴) / 𝑦]𝜑[(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑))
108, 9syl 14 . 2 (𝐴 = ⟨𝑥, 𝑦⟩ → ([(2nd𝐴) / 𝑦]𝜑[(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑))
116, 10bitr2d 188 1 (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st𝐴) / 𝑥][(2nd𝐴) / 𝑦]𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1348  [wsbc 2955  cop 3584  cfv 5196  1st c1st 6114  2nd c2nd 6115
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fv 5204  df-1st 6116  df-2nd 6117
This theorem is referenced by:  dfopab2  6165  dfoprab3s  6166
  Copyright terms: Public domain W3C validator