Theorem adh-minimp-imim1

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

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

Proof

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