Theorem 3imp3i2an 1364

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		3imp3i2an.2	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜏)
3	2	3adant2 1149	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)
4		3imp3i2an.3	. 2 ⊢ ((𝜃 ∧ 𝜏) → 𝜂)
5	1, 3, 4	syl2anc 596	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  focofo  6809  ordunel  7829  naddel1  8680  distrlem5pr  11027  divmul  11890  modmulnn  13940  modaddid  13961  moddi  13993  repswpfx  14846  shftval2  15136  pcgcd  16960  gsumccat  18937  qussub  19306  gsumdixp  20446  lspun  21158  evlslem4  22277  ordtcld3  23406  leadds1im  28231  fusgrfisstep  29737  cplgr3v  29843  upgr2pthnlp  30145  frgrreg  30816  eliuniin  45875  eliuniin2  45896  disjinfi  45968