Theorem re1tbw1

Description: tbw-ax1 rederived from merco2 . (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	re1tbw1	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mercolem8	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) )
2		mercolem3	⊢ ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )
3		mercolem6	⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) )
4	1 2 3	mpsyl	⊢ ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
5		mercolem6	⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
6	4 5	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )