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Theorem merco1lem16 1697
 Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem16 (((𝜑 → (𝜓𝜒)) → 𝜏) → ((𝜑𝜒) → 𝜏))

Proof of Theorem merco1lem16
StepHypRef Expression
1 merco1lem15 1696 . . 3 ((𝜑𝜒) → (𝜑 → (𝜓𝜒)))
2 merco1lem11 1692 . . 3 (((𝜑𝜒) → (𝜑 → (𝜓𝜒))) → ((((𝜏𝜑) → ((𝜑𝜒) → ⊥)) → ⊥) → (𝜑 → (𝜓𝜒))))
31, 2ax-mp 5 . 2 ((((𝜏𝜑) → ((𝜑𝜒) → ⊥)) → ⊥) → (𝜑 → (𝜓𝜒)))
4 merco1 1678 . 2 (((((𝜏𝜑) → ((𝜑𝜒) → ⊥)) → ⊥) → (𝜑 → (𝜓𝜒))) → (((𝜑 → (𝜓𝜒)) → 𝜏) → ((𝜑𝜒) → 𝜏)))
53, 4ax-mp 5 1 (((𝜑 → (𝜓𝜒)) → 𝜏) → ((𝜑𝜒) → 𝜏))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ⊥wfal 1528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 197  df-tru 1526  df-fal 1529 This theorem is referenced by:  merco1lem17  1698  retbwax1  1700
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