Theorem mpani 434

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 433	1 ⊢ (𝜑 → (𝜒 → 𝜃))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  mp2ani  436  mulgt1  9196  recgt1i  9231  recreclt  9233  nngt0  9332  nnrecgt0  9345  elnnnn0c  9613  elnnz1  9672  recnz  9744  uz3m2nn  9983  ledivge1le  10138  expubnd  11047  expnbnd  11115  expnlbnd  11116  sin02gt0  12549  oddge22np1  12666  dvdsnprmd  12921  prmlem1  13244  prmlem2  13256  reeff1olem  15924  sinq12gt0  15984  logdivlti  16036  gausslemma2dlem4  16305