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  5560  omeulem1  8549  divmuldiv  11889  imasmnd2  18708  imasgrp2  18994  imasrng  20093  srgbinomlem2  20143  imasring  20246  abvdiv  20745  mdetunilem9  22514  lly1stc  23390  icccvx  24855  dchrpt  27185  dipsubdir  30784  poimirlem4  37625  fdc  37746  unichnidl  38032  dmncan1  38077  pexmidlem6N  39976  erngdvlem3  40991  erngdvlem3-rN  40999  dvalveclem  41026  dvhvaddass  41098  dvhlveclem  41109  issmflem  46732  prproropf1olem3  47510