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  2866  ovig  6040  prcdnql  7544  prcunqu  7545  p1le  8868  nnge1  9005  zltp1le  9371  gtndiv  9412  uzss  9613  addlelt  9834  xrre2  9887  xrre3  9888  zltaddlt1le  10073  modfzo0difsn  10466  seqf1oglem1  10590  leexp2r  10664  expnlbnd2  10736  facavg  10817  wrdred1hash  10957  caubnd2  11261  maxleast  11357  mulcn2  11455  cn1lem  11457  climsqz  11478  climsqz2  11479  climcvg1nlem  11492  fsumabs  11608  cvgratnnlemnexp  11667  cvgratnnlemmn  11668  zsupcllemstep  12082  gcdzeq  12159  algcvgblem  12187  algcvga  12189  lcmdvdsb  12222  coprm  12282  pclemub  12425  bldisj  14569  xblm  14585  metss2lem  14665  bdxmet  14669  limccoap  14832  lgsne0  15154  gausslemma2dlem1a  15174