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Theorem merco2 1651
Description: A single axiom for propositional calculus offered by Meredith.

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1628. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
merco2 (((𝜑𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃𝜑) → (𝜏 → (𝜂𝜑))))

Proof of Theorem merco2
StepHypRef Expression
1 falim 1488 . . . . . 6 (⊥ → 𝜒)
2 pm2.04 87 . . . . . 6 (((𝜑𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((⊥ → 𝜒) → ((𝜑𝜓) → 𝜃)))
31, 2mpi 20 . . . . 5 (((𝜑𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜑𝜓) → 𝜃))
4 jarl 173 . . . . . 6 (((𝜑𝜓) → 𝜃) → (¬ 𝜑𝜃))
5 idd 24 . . . . . 6 (((𝜑𝜓) → 𝜃) → (𝜃𝜃))
64, 5jad 172 . . . . 5 (((𝜑𝜓) → 𝜃) → ((𝜑𝜃) → 𝜃))
7 looinv 192 . . . . 5 (((𝜑𝜃) → 𝜃) → ((𝜃𝜑) → 𝜑))
83, 6, 73syl 18 . . . 4 (((𝜑𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃𝜑) → 𝜑))
98a1dd 47 . . 3 (((𝜑𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃𝜑) → (𝜏𝜑)))
109a1i 11 . 2 (𝜂 → (((𝜑𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃𝜑) → (𝜏𝜑))))
1110com4l 89 1 (((𝜑𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃𝜑) → (𝜏 → (𝜂𝜑))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1479
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 195  df-tru 1477  df-fal 1480
This theorem is referenced by:  mercolem1  1652  mercolem2  1653  mercolem3  1654  mercolem4  1655  mercolem5  1656  mercolem6  1657  mercolem7  1658  mercolem8  1659  re1tbw4  1663
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