Theorem mpand 429

Description: A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Hypotheses

Ref	Expression
mpand.1	⊢ (𝜑 → 𝜓)
mpand.2	⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))

Assertion

Ref	Expression
mpand	⊢ (𝜑 → (𝜒 → 𝜃))

Proof of Theorem mpand

Step	Hyp	Ref	Expression
1		mpand.1	. 2 ⊢ (𝜑 → 𝜓)
2		mpand.2	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
3	2	ancomsd 269	. 2 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) → 𝜃))
4	1, 3	mpan2d 428	1 ⊢ (𝜑 → (𝜒 → 𝜃))

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 depends on definitions:  df-bi 117

This theorem is referenced by:  mpani  430  mp2and  433  rspcimedv  2870  ovig  6048  prcdnql  7570  prcunqu  7571  p1le  8895  nnge1  9032  zltp1le  9399  gtndiv  9440  uzss  9641  addlelt  9862  xrre2  9915  xrre3  9916  zltaddlt1le  10101  zsupcllemstep  10338  modfzo0difsn  10506  seqf1oglem1  10630  leexp2r  10704  expnlbnd2  10776  facavg  10857  wrdred1hash  10997  caubnd2  11301  maxleast  11397  mulcn2  11496  cn1lem  11498  climsqz  11519  climsqz2  11520  climcvg1nlem  11533  fsumabs  11649  cvgratnnlemnexp  11708  cvgratnnlemmn  11709  bitsfzolem  12138  bitsfzo  12139  gcdzeq  12216  algcvgblem  12244  algcvga  12246  lcmdvdsb  12279  coprm  12339  pclemub  12483  bldisj  14723  xblm  14739  metss2lem  14819  bdxmet  14823  limccoap  15000  lgsne0  15365  gausslemma2dlem1a  15385