Theorem mpand 433

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 432	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  434  mp2and  437  rspcimedv  2931  ovig  6204  prcdnql  7845  prcunqu  7846  p1le  9173  nnge1  9310  zltp1le  9682  gtndiv  9724  uzss  9926  addlelt  10152  xrre2  10206  xrre3  10207  zltaddlt1le  10393  nn0p1elfzo  10577  zsupcllemstep  10645  modfzo0difsn  10815  seqf1oglem1  10939  leexp2r  11013  expnlbnd2  11086  facavg  11167  wrdred1hash  11331  ccat2s1fvwd  11398  caubnd2  11866  maxleast  11962  mulcn2  12061  cn1lem  12063  climsqz  12084  climsqz2  12085  climcvg1nlem  12098  fsumabs  12215  cvgratnnlemnexp  12274  cvgratnnlemmn  12275  bitsfzolem  12704  bitsfzo  12705  gcdzeq  12782  algcvgblem  12810  algcvga  12812  lcmdvdsb  12845  coprm  12905  pclemub  13049  bldisj  15485  xblm  15501  metss2lem  15581  bdxmet  15585  limccoap  15762  lgsne0  16140  gausslemma2dlem1a  16160  eupth2lemsfi  16702