Theorem merco2 1743

Description: A single axiom for propositional calculus discovered by C. A. Meredith.

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1720. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco2	⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃 → 𝜑) → (𝜏 → (𝜂 → 𝜑))))

Proof of Theorem merco2

Step	Hyp	Ref	Expression
1		falim 1559	. . . . . 6 ⊢ (⊥ → 𝜒)
2		pm2.04 90	. . . . . 6 ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((⊥ → 𝜒) → ((𝜑 → 𝜓) → 𝜃)))
3	1, 2	mpi 20	. . . . 5 ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜑 → 𝜓) → 𝜃))
4		jarl 125	. . . . . 6 ⊢ (((𝜑 → 𝜓) → 𝜃) → (¬ 𝜑 → 𝜃))
5		idd 24	. . . . . 6 ⊢ (((𝜑 → 𝜓) → 𝜃) → (𝜃 → 𝜃))
6	4, 5	jad 190	. . . . 5 ⊢ (((𝜑 → 𝜓) → 𝜃) → ((𝜑 → 𝜃) → 𝜃))
7		looinv 206	. . . . 5 ⊢ (((𝜑 → 𝜃) → 𝜃) → ((𝜃 → 𝜑) → 𝜑))
8	3, 6, 7	3syl 18	. . . 4 ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃 → 𝜑) → 𝜑))
9	8	a1dd 50	. . 3 ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃 → 𝜑) → (𝜏 → 𝜑)))
10	9	a1i 11	. 2 ⊢ (𝜂 → (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃 → 𝜑) → (𝜏 → 𝜑))))
11	10	com4l 92	1 ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃 → 𝜑) → (𝜏 → (𝜂 → 𝜑))))

Syntax hints:   → wi 4  ⊥wfal 1554

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 210  df-tru 1545  df-fal 1555

This theorem is referenced by:  mercolem1  1744  mercolem2  1745  mercolem3  1746  mercolem4  1747  mercolem5  1748  mercolem6  1749  mercolem7  1750  mercolem8  1751  re1tbw4  1755