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		mercolem3	⊢ ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )
2		mercolem8	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 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	2 3	ax-mp	⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
5		mercolem3	⊢ ( ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) )
6	4 5	ax-mp	⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) )
7		mercolem8	⊢ ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) )
8		mercolem6	⊢ ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) → ( ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) )
9	7 8	ax-mp	⊢ ( ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) ) )
10	6 9	ax-mp	⊢ ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) )
11	1 10	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) )
12		mercolem6	⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) )
13	11 12	ax-mp	⊢ ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
14		mercolem6	⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
15	13 14	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )