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  9671  prmodvdslcmf  16994  monmat2matmon  22687  cnextcn  23930  umgredg  29041  crctcshwlkn0lem5  29717  tpr2rico  33875  bj-snsetex  36924  bj-restuni  37058  poimirlem26  37613  seqpo  37714  isdrngo2  37925  pm10.55  44331  2pm13.193VD  44865  gpgedg2iv  48031