Theorem imdistanri 723

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 649	1 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 383

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 196  df-an 385

This theorem is referenced by:  tc2  8501  prmodvdslcmf  15589  monmat2matmon  20448  cnextcn  21681  umgredg  25812  usgrarnedg  25913  tpr2rico  29286  bj-snsetex  32144  bj-restuni  32231  poimirlem26  32605  seqpo  32713  isdrngo2  32927  pm10.55  37590  2pm13.193VD  38161  crctcsh1wlkn0lem5  41017