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  7855  mullocpr  7939  lt2halves  9546  nn0n0n1ge2  9720  ixxssixx  10315  pfxsuffeqwrdeq  11486  pfxccatpfx1  11524  pfxccatpfx2  11525  sumtp  12200  dvdsmulcr  12607  dvds2add  12611  dvds2sub  12612  dvdstr  12614  dfgrp3me  13958  uhgrissubgr  16668  subgrprop3  16669  0uhgrsubgr  16672  wlkex  16732  wlkelwrd  16760  bj-peano4  17147