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  9741  prmodvdslcmf  17225  isdrng3lem2  21006  monmat2matmon  23142  cnextcn  24386  umgredg  29716  crctcshwlkn0lem5  30403  tpr2rico  34544  axtco2g  37265  bj-snsetex  37876  bj-restuni  38018  poimirlem26  38565  seqpo  38682  isdrngo2  38893  pm10.55  45353  2pm13.193VD  45885  gpgedg2iv  49165