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Theorem imim21b 356
 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 bi2.04 350 . 2 (((φχ) → (ψθ)) ↔ (ψ → ((φχ) → θ)))
2 pm5.5 326 . . . . 5 (φ → ((φχ) ↔ χ))
32imbi1d 308 . . . 4 (φ → (((φχ) → θ) ↔ (χθ)))
43imim2i 13 . . 3 ((ψφ) → (ψ → (((φχ) → θ) ↔ (χθ))))
54pm5.74d 238 . 2 ((ψφ) → ((ψ → ((φχ) → θ)) ↔ (ψ → (χθ))))
61, 5syl5bb 248 1 ((ψφ) → (((φχ) → (ψθ)) ↔ (ψ → (χθ))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177 This theorem is referenced by: (None)
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