Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  imim21b GIF version

Theorem imim21b 245
 Description: Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Wolf Lammen, 14-Sep-2013.)
Assertion
Ref Expression
imim21b ((𝜓𝜑) → (((𝜑𝜒) → (𝜓𝜃)) ↔ (𝜓 → (𝜒𝜃))))

Proof of Theorem imim21b
StepHypRef Expression
1 bi2.04 241 . 2 (((𝜑𝜒) → (𝜓𝜃)) ↔ (𝜓 → ((𝜑𝜒) → 𝜃)))
2 pm5.5 235 . . . . 5 (𝜑 → ((𝜑𝜒) ↔ 𝜒))
32imbi1d 224 . . . 4 (𝜑 → (((𝜑𝜒) → 𝜃) ↔ (𝜒𝜃)))
43imim2i 12 . . 3 ((𝜓𝜑) → (𝜓 → (((𝜑𝜒) → 𝜃) ↔ (𝜒𝜃))))
54pm5.74d 175 . 2 ((𝜓𝜑) → ((𝜓 → ((𝜑𝜒) → 𝜃)) ↔ (𝜓 → (𝜒𝜃))))
61, 5syl5bb 185 1 ((𝜓𝜑) → (((𝜑𝜒) → (𝜓𝜃)) ↔ (𝜓 → (𝜒𝜃))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105 This theorem depends on definitions:  df-bi 114 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator