Theorem mp3an23 1269

Description: An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)

Hypotheses

Ref	Expression
mp3an23.1	⊢ ψ
mp3an23.2	⊢ χ
mp3an23.3	⊢ ((φ ∧ ψ ∧ χ) → θ)

Assertion

Ref	Expression
mp3an23	⊢ (φ → θ)

Proof of Theorem mp3an23

Step	Hyp	Ref	Expression
1		mp3an23.1	. 2 ⊢ ψ
2		mp3an23.2	. . 3 ⊢ χ
3		mp3an23.3	. . 3 ⊢ ((φ ∧ ψ ∧ χ) → θ)
4	2, 3	mp3an3 1266	. 2 ⊢ ((φ ∧ ψ) → θ)
5	1, 4	mpan2 652	1 ⊢ (φ → 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:  sbciegf  3078  vfintle  4547  vfin1cltv  4548  funpr  5152  enmap2lem5  6068  enprmaplem5  6081  nchoicelem17  6306