Theorem mpan12 530

Description: An inference based on modus ponens.

Hypotheses

Ref	Expression
mpan12.1	⊢ ψ
mpan12.2	⊢ (((φ ∧ ψ) ∧ χ) → θ)

Assertion

Ref	Expression
mpan12	⊢ ((φ ∧ χ) → θ)

Proof of Theorem mpan12

Step	Hyp	Ref	Expression
1		mpan12.1	. . 3 ⊢ ψ
2		mpan12.2	. . . 4 ⊢ (((φ ∧ ψ) ∧ χ) → θ)
3	2	exp 291	. . 3 ⊢ ((φ ∧ ψ) → (χ → θ))
4	1, 3	mpan2 519	. 2 ⊢ (φ → (χ → θ))
5	4	imp 277	1 ⊢ ((φ ∧ χ) → θ)

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  mp3an2 640  limom 2387  tfrlem11 2959  tfr3 2964  oe0 3130  infensuc 3484  ac6lem 3575  indpi 3828  prlem934b 3932  axcnre 4087  om2uzran 4655

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

metamath.org