Theorem imim21b 394

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		bi2.04 388	. 2 ⊢ (((𝜑 → 𝜒) → (𝜓 → 𝜃)) ↔ (𝜓 → ((𝜑 → 𝜒) → 𝜃)))
2		pm5.5 361	. . . . 5 ⊢ (𝜑 → ((𝜑 → 𝜒) ↔ 𝜒))
3	2	imbi1d 341	. . . 4 ⊢ (𝜑 → (((𝜑 → 𝜒) → 𝜃) ↔ (𝜒 → 𝜃)))
4	3	imim2i 16	. . 3 ⊢ ((𝜓 → 𝜑) → (𝜓 → (((𝜑 → 𝜒) → 𝜃) ↔ (𝜒 → 𝜃))))
5	4	pm5.74d 272	. 2 ⊢ ((𝜓 → 𝜑) → ((𝜓 → ((𝜑 → 𝜒) → 𝜃)) ↔ (𝜓 → (𝜒 → 𝜃))))
6	1, 5	syl5bb 282	1 ⊢ ((𝜓 → 𝜑) → (((𝜑 → 𝜒) → (𝜓 → 𝜃)) ↔ (𝜓 → (𝜒 → 𝜃))))

Syntax hints:   → wi 4   ↔ wb 205

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 206

This theorem is referenced by: (None)