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	⊢ ((ψ → φ) → (((φ → χ) → (ψ → θ)) ↔ (ψ → (χ → θ))))

Proof of Theorem imim21b

Step	Hyp	Ref	Expression
1		bi2.04 350	. 2 ⊢ (((φ → χ) → (ψ → θ)) ↔ (ψ → ((φ → χ) → θ)))
2		pm5.5 326	. . . . 5 ⊢ (φ → ((φ → χ) ↔ χ))
3	2	imbi1d 308	. . . 4 ⊢ (φ → (((φ → χ) → θ) ↔ (χ → θ)))
4	3	imim2i 13	. . 3 ⊢ ((ψ → φ) → (ψ → (((φ → χ) → θ) ↔ (χ → θ))))
5	4	pm5.74d 238	. 2 ⊢ ((ψ → φ) → ((ψ → ((φ → χ) → θ)) ↔ (ψ → (χ → θ))))
6	1, 5	syl5bb 248	1 ⊢ ((ψ → φ) → (((φ → χ) → (ψ → θ)) ↔ (ψ → (χ → θ))))

Syntax hints:   → wi 4   ↔ wb 176

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 177

This theorem is referenced by: (None)