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  9574  dfom3  9600  dfac5lem4  10079  dfac5lem4OLD  10081  dfac9  10090  cflem  10198  cflemOLD  10199  canthp1lem2  10606  addsrpr  11028  mulsrpr  11029  trclublem  14961  gcdaddmlem  16494  tgjustf  28400  sto1i  32165  stji1i  32171  kur14lem9  35201  dfon2lem4  35774  rtrclex  43606  comptiunov2i  43695