Theorem simpr31 1282

Description: Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.)

Assertion

Ref	Expression
simpr31	⊢ ((𝜂 ∧ (𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜑)

Proof of Theorem simpr31

Step	Hyp	Ref	Expression
1		simpr1 1213	. 2 ⊢ ((𝜂 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒)) → 𝜑)
2	1	3ad2antr3 1209	1 ⊢ ((𝜂 ∧ (𝜃 ∧ 𝜏 ∧ (𝜑 ∧ 𝜓 ∧ 𝜒))) → 𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  oppccatid  17898  subccatid  18026  fuccatid  18152  setccatid  18264  catccatid  18286  estrccatid  18311  xpccatid  18367  nllyidm  23813  utoptop  24558  cgr3tr4  36909  seglecgr12im  36967  paddasslem9  40978  cdlemd1  41348  cdlemf2  41712  cdlemk34  42060  dihmeetlem18N  42474  dihmeetlem19N  42475  ssccatid  50279  isthincd2  50665  mndtccatid  50818