Theorem re1tbw3

Description: tbw-ax3 rederived from merco2 . (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	re1tbw3	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		mercolem2	⊢ ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) )
2		mercolem2	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) ) )
3		mercolem6	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) )
5	1 4	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 )