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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400

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 401

This theorem is used by:  tc2  9707  prmodvdslcmf  17113  isdrng3lem2  20863  monmat2matmon  22992  cnextcn  24235  umgredg  29499  crctcshwlkn0lem5  30174  tpr2rico  34311  axtco2g  37016  bj-snsetex  37627  bj-restuni  37767  poimirlem26  38325  seqpo  38426  isdrngo2  38637  pm10.55  45107  2pm13.193VD  45639  gpgedg2iv  48860