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  2843  ovig  5995  prcdnql  7482  prcunqu  7483  p1le  8805  nnge1  8941  zltp1le  9306  gtndiv  9347  uzss  9547  addlelt  9767  xrre2  9820  xrre3  9821  zltaddlt1le  10006  modfzo0difsn  10394  leexp2r  10573  expnlbnd2  10645  facavg  10725  caubnd2  11125  maxleast  11221  mulcn2  11319  cn1lem  11321  climsqz  11342  climsqz2  11343  climcvg1nlem  11356  fsumabs  11472  cvgratnnlemnexp  11531  cvgratnnlemmn  11532  zsupcllemstep  11945  gcdzeq  12022  algcvgblem  12048  algcvga  12050  lcmdvdsb  12083  coprm  12143  pclemub  12286  bldisj  13837  xblm  13853  metss2lem  13933  bdxmet  13937  limccoap  14083  lgsne0  14375