Theorem imdistanri 569

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 552	1 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 395

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

This theorem is referenced by:  tc2  9638  prmodvdslcmf  16959  monmat2matmon  22709  cnextcn  23952  umgredg  29083  crctcshwlkn0lem5  29759  tpr2rico  33879  bj-snsetex  36941  bj-restuni  37075  poimirlem26  37630  seqpo  37731  isdrngo2  37942  pm10.55  44346  2pm13.193VD  44880  gpgedg2iv  48055