Theorem imdistanri 580

Description: Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)

Hypothesis

Ref	Expression
imdistanri.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
imdistanri	⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑))

Proof of Theorem imdistanri

Step	Hyp	Ref	Expression
1		imdistanri.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	com12 33	. 2 ⊢ (𝜓 → (𝜑 → 𝜒))
3	2	impac 562	1 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑))

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

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

This theorem is used by:  tc2  9722  prmodvdslcmf  17143  isdrng3lem2  20916  monmat2matmon  23050  cnextcn  24294  umgredg  29581  crctcshwlkn0lem5  30268  tpr2rico  34409  axtco2g  37083  bj-snsetex  37694  bj-restuni  37834  poimirlem26  38382  seqpo  38484  isdrngo2  38695  pm10.55  45180  2pm13.193VD  45712  gpgedg2iv  48970