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Metamath Proof Explorer

Theorem merco1lem10

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 . (Contributed by Anthony Hart, 18-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merco1lem10 ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜏 → 𝜒 ) ) → 𝜑 ) → ( 𝜃 → 𝜑 ) )

Proof

Step Hyp Ref Expression
1 merco1 ⊢ ( ( ( ( ( 𝜒 → 𝜑 ) → ( 𝜏 → ⊥ ) ) → 𝜑 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜏 → 𝜒 ) ) )
2 merco1lem2 ⊢ ( ( ( ( ( ( 𝜒 → 𝜑 ) → ( 𝜏 → ⊥ ) ) → 𝜑 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜏 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜃 → ⊥ ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( 𝜏 → ⊥ ) ) → 𝜑 ) → ⊥ ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜏 → 𝜒 ) ) ) )
3 1 2 ax-mp ⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜃 → ⊥ ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( 𝜏 → ⊥ ) ) → 𝜑 ) → ⊥ ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜏 → 𝜒 ) ) )
4 merco1 ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜃 → ⊥ ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( 𝜏 → ⊥ ) ) → 𝜑 ) → ⊥ ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜏 → 𝜒 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜏 → 𝜒 ) ) → 𝜑 ) → ( 𝜃 → 𝜑 ) ) )
5 3 4 ax-mp ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜏 → 𝜒 ) ) → 𝜑 ) → ( 𝜃 → 𝜑 ) )