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  2910  ovig  6138  prcdnql  7697  prcunqu  7698  p1le  9022  nnge1  9159  zltp1le  9527  gtndiv  9568  uzss  9770  addlelt  9996  xrre2  10049  xrre3  10050  zltaddlt1le  10235  zsupcllemstep  10482  modfzo0difsn  10650  seqf1oglem1  10774  leexp2r  10848  expnlbnd2  10920  facavg  11001  wrdred1hash  11150  ccat2s1fvwd  11217  caubnd2  11671  maxleast  11767  mulcn2  11866  cn1lem  11868  climsqz  11889  climsqz2  11890  climcvg1nlem  11903  fsumabs  12019  cvgratnnlemnexp  12078  cvgratnnlemmn  12079  bitsfzolem  12508  bitsfzo  12509  gcdzeq  12586  algcvgblem  12614  algcvga  12616  lcmdvdsb  12649  coprm  12709  pclemub  12853  bldisj  15118  xblm  15134  metss2lem  15214  bdxmet  15218  limccoap  15395  lgsne0  15760  gausslemma2dlem1a  15780