Theorem 3imp3i2an 1299

Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 13-Apr-2022.)

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.1	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
2		3simpb 1079	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒))
3		3imp3i2an.2	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜏)
4	2, 3	syl 17	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)
5		3imp3i2an.3	. 2 ⊢ ((𝜃 ∧ 𝜏) → 𝜂)
6	1, 4, 5	syl2anc 694	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054

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 385  df-3an 1056

This theorem is referenced by:  cplgr3v  26387  upgr2pthnlp  26684  frgrreg  27381  eliuniin  39593  eliuniin2  39617