Theorem adh-minimp-imim1 48011

Description: Derivation of imim1 84 ("left antimonotonicity of implication", theorem *2.06 of [WhiteheadRussell] p. 100) from adh-minimp 48005 and ax-mp 5. Polish prefix notation: CCpqCCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
adh-minimp-imim1	⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

Proof of Theorem adh-minimp-imim1

Step	Hyp	Ref	Expression
1		adh-minimp-sylsimp 48009	. 2 ⊢ ((((𝜃 → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
2		adh-minimp-jarr-imim1-ax2c-lem1 48006	. . . 4 ⊢ ((𝜑 → 𝜓) → (((𝜃 → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
3		adh-minimp-jarr-imim1-ax2c-lem1 48006	. . . 4 ⊢ (((𝜑 → 𝜓) → (((𝜃 → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) → (((𝜌 → (𝜑 → 𝜓)) → ((((𝜃 → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))) → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))))
4	2, 3	ax-mp 5	. . 3 ⊢ (((𝜌 → (𝜑 → 𝜓)) → ((((𝜃 → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))) → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))))
5		adh-minimp-sylsimp 48009	. . 3 ⊢ ((((𝜌 → (𝜑 → 𝜓)) → ((((𝜃 → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))) → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))) → (((((𝜃 → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))))
6	4, 5	ax-mp 5	. 2 ⊢ (((((𝜃 → 𝜑) → (𝜓 → 𝜒)) → (𝜑 → 𝜒)) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))))
7	1, 6	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  adh-minimp-ax2c  48012