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 399

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 210  df-an 400

This theorem is referenced by:  inf0  9158  dfom3  9184  dfac5lem4  9627  dfac9  9637  cflem  9747  canthp1lem2  10154  addsrpr  10576  mulsrpr  10577  trclublem  14445  gcdaddmlem  15968  tgjustf  26419  sto1i  30171  stji1i  30177  kur14lem9  32747  dfon2lem4  33334  rtrclex  40762  comptiunov2i  40852