Theorem 3imp3i2an 1346

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 1131	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)
4		3imp3i2an.3	. 2 ⊢ ((𝜃 ∧ 𝜏) → 𝜂)
5	1, 3, 4	syl2anc 584	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086

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 207  df-an 396  df-3an 1088

This theorem is referenced by:  focofo  6803  ordunel  7821  naddel1  8699  distrlem5pr  11041  divmul  11899  modmulnn  13906  moddi  13957  repswpfx  14803  shftval2  15094  pcgcd  16898  gsumccat  18819  qussub  19174  gsumdixp  20279  lspun  20944  evlslem4  22034  ordtcld3  23137  sleadd1im  27946  fusgrfisstep  29308  cplgr3v  29414  upgr2pthnlp  29714  frgrreg  30375  eliuniin  45123  eliuniin2  45144  disjinfi  45216