Theorem imdistani 443

Description: Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)

Hypothesis

Ref	Expression
imdistani.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
imdistani	⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))

Proof of Theorem imdistani

Step	Hyp	Ref	Expression
1		imdistani.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	anc2li 327	. 2 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒)))
3	2	imp 123	1 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem is referenced by:  xoranor  1372  nfan1  1557  sbcof2  1803  difin  3364  difrab  3401  opthreg  4540  wessep  4562  fvelimab  5552  elfvmptrab  5591  dffo4  5644  dffo5  5645  ltaddpr  7559  recgt1i  8814  elnnnn0c  9180  elnnz1  9235  recnz  9305  eluz2b2  9562  elfzp12  10055  cos01gt0  11725  oddnn02np1  11839  reumodprminv  12207  sgrpidmndm  12656  bj-charfundc  13843