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  5543  omeulem1  8509  divmuldiv  11841  imasmnd2  18699  imasgrp2  18985  imasrng  20112  srgbinomlem2  20162  imasring  20266  abvdiv  20762  mdetunilem9  22564  lly1stc  23440  icccvx  24904  dchrpt  27234  dipsubdir  30923  poimirlem4  37825  fdc  37946  unichnidl  38232  dmncan1  38277  pexmidlem6N  40235  erngdvlem3  41250  erngdvlem3-rN  41258  dvalveclem  41285  dvhvaddass  41357  dvhlveclem  41368  issmflem  46971  prproropf1olem3  47751