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Theorem merco1lem6 1486
 Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1478. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem6 ((φ → (φψ)) → (χ → (φψ)))

Proof of Theorem merco1lem6
StepHypRef Expression
1 merco1lem5 1485 . . . . 5 (((((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ ) → ⊥ ) → ((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ))
2 merco1lem3 1483 . . . . 5 ((((((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ ) → ⊥ ) → ((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ )) → ((((φψ) → ⊥ ) → (χ → ⊥ )) → (((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ )))
31, 2ax-mp 8 . . . 4 ((((φψ) → ⊥ ) → (χ → ⊥ )) → (((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ ))
4 merco1lem5 1485 . . . 4 (((((φψ) → ⊥ ) → (χ → ⊥ )) → (((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ )) → ((φψ) → (((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ )))
53, 4ax-mp 8 . . 3 ((φψ) → (((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ ))
6 merco1lem3 1483 . . 3 (((φψ) → (((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ )) → (((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → φ))
75, 6ax-mp 8 . 2 (((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → φ)
8 merco1 1478 . 2 ((((((φψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → φ) → ((φ → (φψ)) → (χ → (φψ))))
97, 8ax-mp 8 1 ((φ → (φψ)) → (χ → (φψ)))
 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:  merco1lem7  1487  merco1lem8  1489
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