Theorem mpani 430

Description: An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)

Hypotheses

Ref	Expression
mpani.1	⊢ 𝜓
mpani.2	⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))

Assertion

Ref	Expression
mpani	⊢ (𝜑 → (𝜒 → 𝜃))

Proof of Theorem mpani

Step	Hyp	Ref	Expression
1		mpani.1	. . 3 ⊢ 𝜓
2	1	a1i 9	. 2 ⊢ (𝜑 → 𝜓)
3		mpani.2	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
4	2, 3	mpand 429	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:  mp2ani  432  mulgt1  8909  recgt1i  8944  recreclt  8946  nngt0  9034  nnrecgt0  9047  elnnnn0c  9313  elnnz1  9368  recnz  9438  uz3m2nn  9666  ledivge1le  9820  expubnd  10707  expnbnd  10774  expnlbnd  10775  sin02gt0  11948  oddge22np1  12065  dvdsnprmd  12320  reeff1olem  15115  sinq12gt0  15174  logdivlti  15225  gausslemma2dlem4  15413