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Metamath Proof Explorer

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 ) ) )