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

Theorem csbiedf 2968
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbiedf.1 𝑥𝜑
csbiedf.2 (𝜑𝑥𝐶)
csbiedf.3 (𝜑𝐴𝑉)
csbiedf.4 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
csbiedf (𝜑𝐴 / 𝑥𝐵 = 𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbiedf
StepHypRef Expression
1 csbiedf.1 . . 3 𝑥𝜑
2 csbiedf.4 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
32ex 113 . . 3 (𝜑 → (𝑥 = 𝐴𝐵 = 𝐶))
41, 3alrimi 1460 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶))
5 csbiedf.3 . . 3 (𝜑𝐴𝑉)
6 csbiedf.2 . . 3 (𝜑𝑥𝐶)
7 csbiebt 2967 . . 3 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
85, 6, 7syl2anc 403 . 2 (𝜑 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
94, 8mpbid 145 1 (𝜑𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wnf 1394  wcel 1438  wnfc 2215  csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by:  csbied  2974  csbie2t  2976
  Copyright terms: Public domain W3C validator