This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )