Theorem mpani 521

Description: An inference based on modus ponens.

Hypotheses

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

Assertion

Ref	Expression
mpani	⊢ (φ → (χ → θ))

Proof of Theorem mpani

Step	Hyp	Ref	Expression
1		mpani.1	. 2 ⊢ ψ
2		mpani.2	. . 3 ⊢ (φ → ((ψ ∧ χ) → θ))
3	2	exp3a 292	. 2 ⊢ (φ → (ψ → (χ → θ)))
4	1, 3	mpi 44	1 ⊢ (φ → (χ → θ))

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  mp2ani 523  mpan21 531  onfr 2237  nngt0t 4441  nnrecgt0t 4447  znnen 4930  shsubclt 5125  dmdbr2 5733

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