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

Theorem sblim 1828
 Description: Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sblim.1 xψ
Assertion
Ref Expression
sblim ([y / x](φψ) ↔ ([y / x]φψ))

Proof of Theorem sblim
StepHypRef Expression
1 sbim 1824 . 2 ([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))
2 sblim.1 . . . 4 xψ
32sbf 1657 . . 3 ([y / x]ψψ)
43imbi2i 215 . 2 (([y / x]φ → [y / x]ψ) ↔ ([y / x]φψ))
51, 4bitri 173 1 ([y / x](φψ) ↔ ([y / x]φψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  Ⅎwnf 1346  [wsb 1642 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by:  sbnf2  1854  sbmo  1956
 Copyright terms: Public domain W3C validator