Theorem 3simpc 1022

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 1009	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒 ∧ 𝜑))
2		3simpa 1020	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → (𝜓 ∧ 𝜒))
3	1, 2	sylbi 121	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004

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 1006

This theorem is referenced by:  simp3  1025  3adant1  1041  3adantl1  1179  3adantr1  1182  eupickb  2160  find  4699  fovcld  6131  fisseneq  7132  eqsupti  7200  divcanap2  8865  diveqap0  8867  divrecap  8873  divcanap3  8883  eliooord  10168  fzrev3  10327  sqdivap  10871  swrdlend  11248  swrdnd  11249  ccats1pfxeqbi  11332  muldvds2  12401  dvdscmul  12402  dvdsmulc  12403  dvdstr  12412  domneq0  14310  znleval2  14692  cncfmptc  15349  cnplimclemr  15422  uhgr2edg  16086  umgr2edgneu  16092  clwwlknp  16297