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  9649  dfom3  9675  dfac5lem4  10232  dfac9  10242  cflem  10350  canthp1lem2  10765  addsrpr  11187  mulsrpr  11188  trclublem  15177  gcdaddmlem  16725  tgjustf  28965  sto1i  32868  stji1i  32874  ontr1i  35789  kur14lem9  36146  dfon2lem4  36726  dfttc3gw  37489  rtrclex  44809  comptiunov2i  44898