Theorem 3adantr1 1170

Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)

Hypothesis

Ref	Expression
3adantr.1	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)

Assertion

Ref	Expression
3adantr1	⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓 ∧ 𝜒)) → 𝜃)

Proof of Theorem 3adantr1

Step	Hyp	Ref	Expression
1		3simpc 1150	. 2 ⊢ ((𝜏 ∧ 𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒))
2		3adantr.1	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
3	1, 2	sylan2 593	1 ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓 ∧ 𝜒)) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086

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 207  df-an 396  df-3an 1088

This theorem is referenced by:  3adant3r1  1183  3ad2antr3  1191  swopo  5557  omeulem1  8546  divmuldiv  11882  imasmnd2  18701  imasgrp2  18987  imasrng  20086  srgbinomlem2  20136  imasring  20239  abvdiv  20738  mdetunilem9  22507  lly1stc  23383  icccvx  24848  dchrpt  27178  dipsubdir  30777  poimirlem4  37618  fdc  37739  unichnidl  38025  dmncan1  38070  pexmidlem6N  39969  erngdvlem3  40984  erngdvlem3-rN  40992  dvalveclem  41019  dvhvaddass  41091  dvhlveclem  41102  issmflem  46725  prproropf1olem3  47506