Theorem imdistanri 577

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 560	1 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 399

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 209  df-an 400

This theorem is referenced by:  tc2  9697  prmodvdslcmf  17085  monmat2matmon  22886  cnextcn  24129  umgredg  29341  crctcshwlkn0lem5  30016  tpr2rico  34211  axtco2g  36842  bj-snsetex  37453  bj-restuni  37592  poimirlem26  38150  seqpo  38251  isdrngo2  38462  pm10.55  44950  2pm13.193VD  45483  gpgedg2iv  48694