Theorem 3imp3i2an 1347

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 1132	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏)
4		3imp3i2an.3	. 2 ⊢ ((𝜃 ∧ 𝜏) → 𝜂)
5	1, 3, 4	syl2anc 585	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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 1089

This theorem is referenced by:  focofo  6765  ordunel  7778  naddel1  8623  distrlem5pr  10950  divmul  11812  modmulnn  13848  modaddid  13869  moddi  13901  repswpfx  14747  shftval2  15037  pcgcd  16849  gsumccat  18809  qussub  19166  gsumdixp  20298  lspun  20982  evlslem4  22054  ordtcld3  23164  leadds1im  27979  fusgrfisstep  29398  cplgr3v  29504  upgr2pthnlp  29800  frgrreg  30464  eliuniin  45529  eliuniin2  45550  disjinfi  45622