Theorem 3imp3i2an 1278

Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)

Hypotheses

Ref	Expression
3imp3i2an.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3imp3i2an.2	⊢ ((𝜑 ∧ 𝜒) → 𝜏)
3imp3i2an.3	⊢ ((𝜃 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
3imp3i2an	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂)

Proof of Theorem 3imp3i2an

Step	Hyp	Ref	Expression
1		3imp3i2an.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜒) → 𝜏)
2		3imp3i2an.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
3	2	3exp 1264	. . . . . . . . 9 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
4		3imp3i2an.3	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝜏) → 𝜂)
5	4	ex 450	. . . . . . . . 9 ⊢ (𝜃 → (𝜏 → 𝜂))
6	3, 5	syl8 76	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂))))
7	6	com4r 94	. . . . . . 7 ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
8	1, 7	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝜒) → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
9	8	ex 450	. . . . 5 ⊢ (𝜑 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
10	9	pm2.43b 55	. . . 4 ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
11	10	com4r 94	. . 3 ⊢ (𝜒 → (𝜒 → (𝜑 → (𝜓 → 𝜂))))
12	11	pm2.43i 52	. 2 ⊢ (𝜒 → (𝜑 → (𝜓 → 𝜂)))
13	12	3imp231 1258	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037

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-an 386  df-3an 1039

This theorem is referenced by:  upgr2pthnlp  26628  frgrreg  27252  eliuniin  39279  eliuniin2  39303