Theorem mp3an12 1267

Description: An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)

Hypotheses

Ref	Expression
mp3an12.1	⊢ φ
mp3an12.2	⊢ ψ
mp3an12.3	⊢ ((φ ∧ ψ ∧ χ) → θ)

Assertion

Ref	Expression
mp3an12	⊢ (χ → θ)

Proof of Theorem mp3an12

Step	Hyp	Ref	Expression
1		mp3an12.2	. 2 ⊢ ψ
2		mp3an12.1	. . 3 ⊢ φ
3		mp3an12.3	. . 3 ⊢ ((φ ∧ ψ ∧ χ) → θ)
4	2, 3	mp3an1 1264	. 2 ⊢ ((ψ ∧ χ) → θ)
5	1, 4	mpan 651	1 ⊢ (χ → θ)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 934

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

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is used by:  ceqsralv  2887  opkelopkabg  4246  otkelins2kg  4254  otkelins3kg  4255  opkelcokg  4262  vfin1cltv  4548  vfinspss  4552  fvfullfunlem3  5864  fvfullfun  5865  clos1nrel  5887  cenc  6182  nclec  6196  nc0le1  6217  nclenc  6223