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 595	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103

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 210  df-an 401  df-3an 1105

This theorem is referenced by:  focofo  6805  ordunel  7819  naddel1  8670  distrlem5pr  11007  divmul  11870  modmulnn  13918  modaddid  13939  moddi  13971  repswpfx  14818  shftval2  15108  pcgcd  16933  gsumccat  18895  qussub  19257  gsumdixp  20396  lspun  21108  evlslem4  22227  ordtcld3  23356  leadds1im  28180  fusgrfisstep  29679  cplgr3v  29785  upgr2pthnlp  30081  frgrreg  30745  eliuniin  45817  eliuniin2  45838  disjinfi  45910