Theorem re1tbw2

Description: tbw-ax2 rederived from merco2 . (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	re1tbw2	⊢ ( 𝜑 → ( 𝜓 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		mercolem1	⊢ ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) )
2		mercolem1	⊢ ( ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜓 → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) ) )
3	1 2	ax-mp	⊢ ( 𝜑 → ( 𝜓 → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) )
4		mercolem6	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) ) → ( 𝜓 → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) )
5	3 4	ax-mp	⊢ ( 𝜓 → ( 𝜑 → ( 𝜓 → 𝜑 ) ) )
6		mercolem6	⊢ ( ( 𝜓 → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) )
7	5 6	ax-mp	⊢ ( 𝜑 → ( 𝜓 → 𝜑 ) )