Theorem mpand 405

Description: A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Hypotheses

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

Assertion

Ref	Expression
mpand	⊢ (φ → (χ → θ))

Proof of Theorem mpand

Step	Hyp	Ref	Expression
1		mpand.1	. 2 ⊢ (φ → ψ)
2		mpand.2	. . 3 ⊢ (φ → ((ψ ∧ χ) → θ))
3	2	ancomsd 256	. 2 ⊢ (φ → ((χ ∧ ψ) → θ))
4	1, 3	mpan2d 404	1 ⊢ (φ → (χ → θ))

Syntax hints:   → wi 4   ∧ wa 97

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  mpani  406  mp2and  409  rspcimedv  2652  ovig  5564  prcdnql  6467  prcunqu  6468  p1le  7596  nnge1  7718  zltp1le  8074  gtndiv  8111  uzss  8269  xrre2  8504  xrre3  8505  leexp2r  8962  expnlbnd2  9027