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Theorem merco1lem11 1492
 Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1478. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem11 ((φψ) → (((χ → (φτ)) → ⊥ ) → ψ))

Proof of Theorem merco1lem11
StepHypRef Expression
1 merco1lem5 1485 . . . . . 6 ((((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ) → ⊥ ) → (((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ))
2 merco1lem3 1483 . . . . . 6 (((((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ) → ⊥ ) → (((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ )) → (((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )))
31, 2ax-mp 8 . . . . 5 (((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ))
4 merco1lem4 1484 . . . . 5 ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )) → ((((χ → (φτ)) → ⊥ ) → ⊥ ) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )))
53, 4ax-mp 8 . . . 4 ((((χ → (φτ)) → ⊥ ) → ⊥ ) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ))
6 merco1lem5 1485 . . . 4 (((((χ → (φτ)) → ⊥ ) → ⊥ ) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )) → ((χ → (φτ)) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )))
75, 6ax-mp 8 . . 3 ((χ → (φτ)) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ))
8 merco1lem4 1484 . . 3 (((χ → (φτ)) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )) → ((φτ) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )))
97, 8ax-mp 8 . 2 ((φτ) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ ))
10 merco1 1478 . . 3 (((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → φ) → ((φψ) → (((χ → (φτ)) → ⊥ ) → ψ)))
11 merco1lem2 1482 . . 3 ((((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → φ) → ((φψ) → (((χ → (φτ)) → ⊥ ) → ψ))) → (((φτ) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )) → ((φψ) → (((χ → (φτ)) → ⊥ ) → ψ))))
1210, 11ax-mp 8 . 2 (((φτ) → ((((ψφ) → (((χ → (φτ)) → ⊥ ) → ⊥ )) → ⊥ ) → ⊥ )) → ((φψ) → (((χ → (φτ)) → ⊥ ) → ψ)))
139, 12ax-mp 8 1 ((φψ) → (((χ → (φτ)) → ⊥ ) → ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊥ wfal 1317 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320 This theorem is referenced by:  merco1lem12  1493  merco1lem16  1497  merco1lem17  1498
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