Theorem mp2ani 699

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 697	. 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  9533  dfom3  9559  dfac5lem4  10039  dfac5lem4OLD  10041  dfac9  10050  cflem  10158  cflemOLD  10159  canthp1lem2  10567  addsrpr  10989  mulsrpr  10990  trclublem  14948  gcdaddmlem  16484  tgjustf  28555  sto1i  32322  stji1i  32328  kur14lem9  35412  dfon2lem4  35982  dfttc3gw  36721  rtrclex  44062  comptiunov2i  44151