ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb6f GIF version

Theorem sb6f 1684
Description: Equivalence for substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.)
Hypothesis
Ref Expression
equs45f.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
sb6f ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb6f
StepHypRef Expression
1 equs45f.1 . . . 4 (𝜑 → ∀𝑦𝜑)
21sbimi 1647 . . 3 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
3 sb4a 1682 . . 3 ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
42, 3syl 14 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
5 sb2 1650 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
64, 5impbii 117 1 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1241  [wsb 1645
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-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-11 1397  ax-4 1400  ax-i9 1423  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-sb 1646
This theorem is referenced by:  sb5f  1685  sbcof2  1691
  Copyright terms: Public domain W3C validator