Theorem imdistani 445

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 329	. 2 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒)))
3	2	imp 124	1 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem is referenced by:  syldanl  449  xoranor  1422  nfan1  1613  sbcof2  1859  difin  3462  difrab  3499  rabsnifsb  3762  opthreg  4683  wessep  4705  fvelimab  5738  elfvmptrab  5778  dffo4  5830  dffo5  5831  ltaddpr  7928  recgt1i  9189  elnnnn0c  9558  elnnz1  9617  recnz  9689  eluz2b2  9953  elfzp12  10455  pfxsuff1eqwrdeq  11416  cos01gt0  12474  oddnn02np1  12591  reumodprminv  12976  ballotfilemfc0  13176  ballotfilemfcc  13177  ballotfilemth  13225  sgrpidmndm  13717  elply2  15712  bj-charfundc  16690