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Theorem imim21b 383
 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 bi2.04 377 . 2 (((𝜑𝜒) → (𝜓𝜃)) ↔ (𝜓 → ((𝜑𝜒) → 𝜃)))
2 pm5.5 352 . . . . 5 (𝜑 → ((𝜑𝜒) ↔ 𝜒))
32imbi1d 332 . . . 4 (𝜑 → (((𝜑𝜒) → 𝜃) ↔ (𝜒𝜃)))
43imim2i 16 . . 3 ((𝜓𝜑) → (𝜓 → (((𝜑𝜒) → 𝜃) ↔ (𝜒𝜃))))
54pm5.74d 264 . 2 ((𝜓𝜑) → ((𝜓 → ((𝜑𝜒) → 𝜃)) ↔ (𝜓 → (𝜒𝜃))))
61, 5syl5bb 274 1 ((𝜓𝜑) → (((𝜑𝜒) → (𝜓𝜃)) ↔ (𝜓 → (𝜒𝜃))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 197 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 198 This theorem is referenced by: (None)
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