Theorem mp3an3 1266

Description: An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)

Hypotheses

Ref	Expression
mp3an3.1	⊢ χ
mp3an3.2	⊢ ((φ ∧ ψ ∧ χ) → θ)

Assertion

Ref	Expression
mp3an3	⊢ ((φ ∧ ψ) → θ)

Proof of Theorem mp3an3

Step	Hyp	Ref	Expression
1		mp3an3.1	. 2 ⊢ χ
2		mp3an3.2	. . 3 ⊢ ((φ ∧ ψ ∧ χ) → θ)
3	2	3expia 1153	. 2 ⊢ ((φ ∧ ψ) → (χ → θ))
4	1, 3	mpi 16	1 ⊢ ((φ ∧ ψ) → θ)

Syntax hints:   → wi 4   ∧ wa 358   ∧ 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:  mp3an13  1268  mp3an23  1269  mp3anl3  1273  vfinnc  4472  ov  5596  ovmpt2a  5715  ovmpt2  5717  enmap1lem5  6074  ltcpw1pwg  6203  nnltp1c  6263  nnc3n3p1  6279