Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbvex4v GIF version

Theorem cbvex4v 1805
 Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 26-Jul-1995.)
Hypotheses
Ref Expression
cbvex4v.1 ((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))
cbvex4v.2 ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))
Assertion
Ref Expression
cbvex4v (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
Distinct variable groups:   𝑧,𝑤,𝜒   𝑣,𝑢,𝜑   𝑥,𝑦,𝜓   𝑓,𝑔,𝜓   𝑤,𝑓   𝑧,𝑔   𝑤,𝑢,𝑥,𝑦,𝑧,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔)   𝜓(𝑧,𝑤,𝑣,𝑢)   𝜒(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)

Proof of Theorem cbvex4v
StepHypRef Expression
1 cbvex4v.1 . . . 4 ((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))
212exbidv 1748 . . 3 ((𝑥 = 𝑣𝑦 = 𝑢) → (∃𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝜓))
32cbvex2v 1799 . 2 (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑧𝑤𝜓)
4 cbvex4v.2 . . . 4 ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))
54cbvex2v 1799 . . 3 (∃𝑧𝑤𝜓 ↔ ∃𝑓𝑔𝜒)
652exbii 1497 . 2 (∃𝑣𝑢𝑧𝑤𝜓 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
73, 6bitri 173 1 (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∃wex 1381 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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by:  enq0sym  6530  addnq0mo  6545  mulnq0mo  6546  addsrmo  6828  mulsrmo  6829
 Copyright terms: Public domain W3C validator