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  5538  omeulem1  8500  divmuldiv  11824  imasmnd2  18648  imasgrp2  18934  imasrng  20062  srgbinomlem2  20112  imasring  20215  abvdiv  20714  mdetunilem9  22505  lly1stc  23381  icccvx  24846  dchrpt  27176  dipsubdir  30792  poimirlem4  37604  fdc  37725  unichnidl  38011  dmncan1  38056  pexmidlem6N  39954  erngdvlem3  40969  erngdvlem3-rN  40977  dvalveclem  41004  dvhvaddass  41076  dvhlveclem  41087  issmflem  46708  prproropf1olem3  47489