Theorem imdistan 670

Description: Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)

Assertion

Ref	Expression
imdistan	⊢ ((φ → (ψ → χ)) ↔ ((φ ∧ ψ) → (φ ∧ χ)))

Proof of Theorem imdistan

Step	Hyp	Ref	Expression
1		pm5.42 531	. 2 ⊢ ((φ → (ψ → χ)) ↔ (φ → (ψ → (φ ∧ χ))))
2		impexp 433	. 2 ⊢ (((φ ∧ ψ) → (φ ∧ χ)) ↔ (φ → (ψ → (φ ∧ χ))))
3	1, 2	bitr4i 243	1 ⊢ ((φ → (ψ → χ)) ↔ ((φ ∧ ψ) → (φ ∧ χ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

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

This theorem is referenced by:  imdistand  673  pm5.3  692  rmoim  3035  ss2rab  3342