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  2909  ovig  6132  prcdnql  7679  prcunqu  7680  p1le  9004  nnge1  9141  zltp1le  9509  gtndiv  9550  uzss  9751  addlelt  9972  xrre2  10025  xrre3  10026  zltaddlt1le  10211  zsupcllemstep  10457  modfzo0difsn  10625  seqf1oglem1  10749  leexp2r  10823  expnlbnd2  10895  facavg  10976  wrdred1hash  11123  caubnd2  11636  maxleast  11732  mulcn2  11831  cn1lem  11833  climsqz  11854  climsqz2  11855  climcvg1nlem  11868  fsumabs  11984  cvgratnnlemnexp  12043  cvgratnnlemmn  12044  bitsfzolem  12473  bitsfzo  12474  gcdzeq  12551  algcvgblem  12579  algcvga  12581  lcmdvdsb  12614  coprm  12674  pclemub  12818  bldisj  15083  xblm  15099  metss2lem  15179  bdxmet  15183  limccoap  15360  lgsne0  15725  gausslemma2dlem1a  15745