Theorem mp2ani 711

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 709	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
5	1, 4	mpi 21	1 ⊢ (𝜑 → 𝜃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  inf0  9600  dfom3  9626  dfac5lem4  10129  dfac9  10139  cflem  10247  canthp1lem2  10656  addsrpr  11078  mulsrpr  11079  trclublem  15058  gcdaddmlem  16607  tgjustf  28779  sto1i  32625  stji1i  32631  kur14lem9  35727  dfon2lem4  36297  dfttc3gw  37075  rtrclex  44384  comptiunov2i  44473