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Metamath Proof Explorer

Theorem retbwax2

Description: tbw-ax2 rederived from merco1 . (Contributed by Anthony Hart, 17-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion retbwax2 ( 𝜑 → ( 𝜓 → 𝜑 ) )

Proof

Step Hyp Ref Expression
1 merco1lem1 ⊢ ( ( ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ⊥ → 𝜑 ) )
2 merco1 ⊢ ( ( ( ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ⊥ → 𝜑 ) ) → ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) )
3 1 2 ax-mp ⊢ ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) )
4 merco1 ⊢ ( ( ( ( ( 𝜑 → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ⊥ ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) )
5 merco1 ⊢ ( ( ( ( ( ( 𝜑 → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ⊥ ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) ) → ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) )
6 4 5 ax-mp ⊢ ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) )
7 3 6 ax-mp ⊢ ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) )
8 merco1lem1 ⊢ ( ( ( ( ( 𝜓 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ⊥ → 𝜑 ) )
9 merco1 ⊢ ( ( ( ( ( ( 𝜓 → 𝜑 ) → 𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ⊥ → 𝜑 ) ) → ( ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) )
10 8 9 ax-mp ⊢ ( ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) )
11 merco1 ⊢ ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜑 ) ) → ( 𝜓 → ⊥ ) ) → ( ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ⊥ ) ) → ⊥ ) → ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) )
12 merco1 ⊢ ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜑 ) ) → ( 𝜓 → ⊥ ) ) → ( ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ⊥ ) ) → ⊥ ) → ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) ) → ( ( ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) → ( ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) ) )
13 11 12 ax-mp ⊢ ( ( ( ( ⊥ → 𝜑 ) → ( 𝜓 → 𝜑 ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) → ( ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) ) )
14 10 13 ax-mp ⊢ ( ( 𝜑 → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) → ( 𝜑 → ( 𝜓 → 𝜑 ) ) )
15 7 14 ax-mp ⊢ ( 𝜑 → ( 𝜓 → 𝜑 ) )