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

Theorem sbimi 1629
Description: Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
Hypothesis
Ref Expression
sbimi.1 (φψ)
Assertion
Ref Expression
sbimi ([y / x]φ → [y / x]ψ)

Proof of Theorem sbimi
StepHypRef Expression
1 sbimi.1 . . . 4 (φψ)
21imim2i 12 . . 3 ((x = yφ) → (x = yψ))
31anim2i 324 . . . 4 ((x = y φ) → (x = y ψ))
43eximi 1473 . . 3 (x(x = y φ) → x(x = y ψ))
52, 4anim12i 321 . 2 (((x = yφ) x(x = y φ)) → ((x = yψ) x(x = y ψ)))
6 df-sb 1628 . 2 ([y / x]φ ↔ ((x = yφ) x(x = y φ)))
7 df-sb 1628 . 2 ([y / x]ψ ↔ ((x = yψ) x(x = y ψ)))
85, 6, 73imtr4i 190 1 ([y / x]φ → [y / x]ψ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wex 1362  [wsb 1627
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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-ial 1409
This theorem depends on definitions:  df-bi 110  df-sb 1628
This theorem is referenced by:  sbbii  1630  sb6f  1666  hbsb3  1671  sbidm  1713  sbco  1824  sbcocom  1826  elsb3  1834  elsb4  1835  sbalyz  1857  hbsb4t  1871  moimv  1948  oprcl  3547  peano1  4244  peano2  4245
  Copyright terms: Public domain W3C validator