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)

Syntax hints:   → wi 4   ∧ w3a 934

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 177  df-an 360  df-3an 936

This theorem is referenced by:  ceqsralv  2886  opkelopkabg  4245  otkelins2kg  4253  otkelins3kg  4254  opkelcokg  4261  vfin1cltv  4547  vfinspss  4551  fvfullfunlem3  5863  fvfullfun  5864  clos1nrel  5886  cenc  6181  nclec  6195  nc0le1  6216  nclenc  6222