Theorem mp2ani 696

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 694	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
5	1, 4	mpi 20	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ∧ wa 398

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 209  df-an 399

This theorem is referenced by:  dfom3  9109  dfac5lem4  9551  dfac9  9561  cflem  9667  canthp1lem2  10074  addsrpr  10496  mulsrpr  10497  trclublem  14354  gcdaddmlem  15871  tgjustf  26258  sto1i  30012  stji1i  30018  kur14lem9  32461  dfon2lem4  33031  rtrclex  39975  comptiunov2i  40049