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  2922  ovig  6174  prcdnql  7795  prcunqu  7796  p1le  9119  nnge1  9256  zltp1le  9628  gtndiv  9669  uzss  9871  addlelt  10097  xrre2  10150  xrre3  10151  zltaddlt1le  10337  nn0p1elfzo  10517  zsupcllemstep  10585  modfzo0difsn  10753  seqf1oglem1  10877  leexp2r  10951  expnlbnd2  11023  facavg  11104  wrdred1hash  11261  ccat2s1fvwd  11328  caubnd2  11795  maxleast  11891  mulcn2  11990  cn1lem  11992  climsqz  12013  climsqz2  12014  climcvg1nlem  12027  fsumabs  12144  cvgratnnlemnexp  12203  cvgratnnlemmn  12204  bitsfzolem  12633  bitsfzo  12634  gcdzeq  12711  algcvgblem  12739  algcvga  12741  lcmdvdsb  12774  coprm  12834  pclemub  12978  bldisj  15253  xblm  15269  metss2lem  15349  bdxmet  15353  limccoap  15530  lgsne0  15898  gausslemma2dlem1a  15918  eupth2lemsfi  16460