Theorem mpan2d 525

Description: A deduction based on modus ponens.

Hypotheses

Ref	Expression
mpan2d.1	⊢ (φ → χ)
mpan2d.2	⊢ (φ → ((ψ ∧ χ) → θ))

Assertion

Ref	Expression
mpan2d	⊢ (φ → (ψ → θ))

Proof of Theorem mpan2d

Step	Hyp	Ref	Expression
1		mpan2d.1	. 2 ⊢ (φ → χ)
2		mpan2d.2	. . 3 ⊢ (φ → ((ψ ∧ χ) → θ))
3	2	exp3a 292	. 2 ⊢ (φ → (ψ → (χ → θ)))
4	1, 3	mpid 48	1 ⊢ (φ → (ψ → θ))

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  alephle 3689  peano2uz 4602  flgzt 4626  shsel1t 5286

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