Theorem 3simpa 1025

Description: Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Assertion

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

Proof of Theorem 3simpa

Step	Hyp	Ref	Expression
1		df-3an 1011	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
2	1	simplbi 274	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  3simpb  1026  3simpc  1027  simp1  1028  simp2  1029  3adant3  1048  3adantl3  1186  3adantr3  1189  opprc  3925  oprcl  3928  opm  4374  funtpg  5432  ftpg  5899  ovig  6210  prltlu  7854  mullocpr  7938  lt2halves  9541  nn0n0n1ge2  9715  ixxssixx  10304  pfxsuffeqwrdeq  11470  pfxccatpfx1  11508  pfxccatpfx2  11509  sumtp  12181  dvdsmulcr  12588  dvds2add  12592  dvds2sub  12593  dvdstr  12595  dfgrp3me  13905  uhgrissubgr  16502  subgrprop3  16503  0uhgrsubgr  16506  wlkex  16566  wlkelwrd  16594  bj-peano4  16981