Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbc3ie GIF version

Theorem sbc3ie 2831
 Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
sbc3ie.1 𝐴 ∈ V
sbc3ie.2 𝐵 ∈ V
sbc3ie.3 𝐶 ∈ V
sbc3ie.4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
Assertion
Ref Expression
sbc3ie ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑𝜓)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐶   𝜓,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem sbc3ie
StepHypRef Expression
1 sbc3ie.1 . 2 𝐴 ∈ V
2 sbc3ie.2 . 2 𝐵 ∈ V
3 sbc3ie.3 . . . 4 𝐶 ∈ V
43a1i 9 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 ∈ V)
5 sbc3ie.4 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
653expa 1104 . . 3 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑧 = 𝐶) → (𝜑𝜓))
74, 6sbcied 2799 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → ([𝐶 / 𝑧]𝜑𝜓))
81, 2, 7sbc2ie 2829 1 ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  Vcvv 2557  [wsbc 2764 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator