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  8503  divmuldiv  11827  imasmnd2  18688  imasgrp2  18974  imasrng  20101  srgbinomlem2  20151  imasring  20254  abvdiv  20750  mdetunilem9  22541  lly1stc  23417  icccvx  24881  dchrpt  27211  dipsubdir  30835  poimirlem4  37670  fdc  37791  unichnidl  38077  dmncan1  38122  pexmidlem6N  40080  erngdvlem3  41095  erngdvlem3-rN  41103  dvalveclem  41130  dvhvaddass  41202  dvhlveclem  41213  issmflem  46830  prproropf1olem3  47610