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  1419  nfan1  1610  sbcof2  1856  difin  3441  difrab  3478  opthreg  4649  wessep  4671  fvelimab  5695  elfvmptrab  5735  dffo4  5788  dffo5  5789  ltaddpr  7800  recgt1i  9061  elnnnn0c  9430  elnnz1  9485  recnz  9556  eluz2b2  9815  elfzp12  10312  pfxsuff1eqwrdeq  11252  cos01gt0  12295  oddnn02np1  12412  reumodprminv  12797  sgrpidmndm  13474  elply2  15430  bj-charfundc  16280