Theorem mp2ani 698

Description: An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)

Hypotheses

Ref	Expression
mp2ani.1	⊢ 𝜓
mp2ani.2	⊢ 𝜒
mp2ani.3	⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))

Assertion

Ref	Expression
mp2ani	⊢ (𝜑 → 𝜃)

Proof of Theorem mp2ani

Step	Hyp	Ref	Expression
1		mp2ani.2	. 2 ⊢ 𝜒
2		mp2ani.1	. . 3 ⊢ 𝜓
3		mp2ani.3	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
4	2, 3	mpani 696	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
5	1, 4	mpi 20	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  inf0  9643  dfom3  9669  dfac5lem4  10148  dfac5lem4OLD  10150  dfac9  10159  cflem  10267  cflemOLD  10268  canthp1lem2  10675  addsrpr  11097  mulsrpr  11098  trclublem  15016  gcdaddmlem  16543  tgjustf  28417  sto1i  32183  stji1i  32189  kur14lem9  35178  dfon2lem4  35746  rtrclex  43592  comptiunov2i  43681