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  8884  recgt1i  8919  recreclt  8921  nngt0  9009  nnrecgt0  9022  elnnnn0c  9288  elnnz1  9343  recnz  9413  uz3m2nn  9641  ledivge1le  9795  expubnd  10670  expnbnd  10737  expnlbnd  10738  sin02gt0  11910  oddge22np1  12025  dvdsnprmd  12266  reeff1olem  14947  sinq12gt0  15006  logdivlti  15057  gausslemma2dlem4  15221