Theorem 3simpc 1020

Description: Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Assertion

Ref	Expression
3simpc	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒))

Proof of Theorem 3simpc

Step	Hyp	Ref	Expression
1		3anrot 1007	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒 ∧ 𝜑))
2		3simpa 1018	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → (𝜓 ∧ 𝜒))
3	1, 2	sylbi 121	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 1004

This theorem is referenced by:  simp3  1023  3adant1  1039  3adantl1  1177  3adantr1  1180  eupickb  2159  find  4692  fovcld  6118  fisseneq  7112  eqsupti  7179  divcanap2  8843  diveqap0  8845  divrecap  8851  divcanap3  8861  eliooord  10141  fzrev3  10300  sqdivap  10842  swrdlend  11211  swrdnd  11212  ccats1pfxeqbi  11295  muldvds2  12349  dvdscmul  12350  dvdsmulc  12351  dvdstr  12360  domneq0  14257  znleval2  14639  cncfmptc  15291  cnplimclemr  15364  uhgr2edg  16025  umgr2edgneu  16031  clwwlknp  16185