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

Theorem sbrim 1873
 Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
sbrim.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
sbrim ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem sbrim
StepHypRef Expression
1 sbim 1870 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sbrim.1 . . . 4 (𝜑 → ∀𝑥𝜑)
32sbh 1701 . . 3 ([𝑦 / 𝑥]𝜑𝜑)
43imbi1i 236 . 2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
51, 4bitri 182 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1283  [wsb 1687 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469 This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688 This theorem is referenced by:  sbco2d  1883  sbco2vd  1884  hbsbd  1901
 Copyright terms: Public domain W3C validator