Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbie2g GIF version

Theorem csbie2g 3050
 Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2943 avoids a disjointness condition on 𝑥 and 𝐴 by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
Hypotheses
Ref Expression
csbie2g.1 (𝑥 = 𝑦𝐵 = 𝐶)
csbie2g.2 (𝑦 = 𝐴𝐶 = 𝐷)
Assertion
Ref Expression
csbie2g (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐷)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶   𝑦,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem csbie2g
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3004 . 2 𝐴 / 𝑥𝐵 = {𝑧[𝐴 / 𝑥]𝑧𝐵}
2 csbie2g.1 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
32eleq2d 2209 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
4 csbie2g.2 . . . . 5 (𝑦 = 𝐴𝐶 = 𝐷)
54eleq2d 2209 . . . 4 (𝑦 = 𝐴 → (𝑧𝐶𝑧𝐷))
63, 5sbcie2g 2942 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐵𝑧𝐷))
76abbi1dv 2259 . 2 (𝐴𝑉 → {𝑧[𝐴 / 𝑥]𝑧𝐵} = 𝐷)
81, 7syl5eq 2184 1 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  {cab 2125  [wsbc 2909  ⦋csb 3003 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator