Theorem imdistanri 579

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 561	1 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 400

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 210  df-an 401

This theorem is referenced by:  tc2  9708  prmodvdslcmf  17106  monmat2matmon  22960  cnextcn  24203  umgredg  29454  crctcshwlkn0lem5  30129  tpr2rico  34268  axtco2g  36932  bj-snsetex  37543  bj-restuni  37683  poimirlem26  38241  seqpo  38342  isdrngo2  38553  pm10.55  45027  2pm13.193VD  45559  gpgedg2iv  48777