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

Theorem sbiedv 1650
 Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1652). (Contributed by NM, 7-Jan-2017.)
Hypothesis
Ref Expression
sbiedv.1 ((φ x = y) → (ψχ))
Assertion
Ref Expression
sbiedv (φ → ([y / x]ψχ))
Distinct variable groups:   φ,x   χ,x
Allowed substitution hints:   φ(y)   ψ(x,y)   χ(y)

Proof of Theorem sbiedv
StepHypRef Expression
1 nfv 1398 . 2 xφ
2 nfvd 1399 . 2 (φ → Ⅎxχ)
3 sbiedv.1 . . 3 ((φ x = y) → (ψχ))
43ex 108 . 2 (φ → (x = y → (ψχ)))
51, 2, 4sbied 1649 1 (φ → ([y / x]ψχ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  [wsb 1623 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 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624 This theorem is referenced by:  acexmid  5431
 Copyright terms: Public domain W3C validator