Theorem imdistanri 571

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 32	. 2 ⊢ (𝜓 → (𝜑 → 𝜒))
3	2	impac 554	1 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 397

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 206  df-an 398

This theorem is referenced by:  tc2  9733  prmodvdslcmf  16976  monmat2matmon  22308  cnextcn  23553  umgredg  28378  crctcshwlkn0lem5  29048  tpr2rico  32830  bj-snsetex  35782  bj-restuni  35916  poimirlem26  36452  seqpo  36553  isdrngo2  36764  pm10.55  43061  2pm13.193VD  43597