Theorem msca 508

Description: Syllogism combined with contraposition.

Hypotheses

Ref	Expression
msca.1	⊢ (φ → (ψ → χ))
msca.2	⊢ (θ → (ψ → ¬ χ))

Assertion

Ref	Expression
msca	⊢ (φ → (ψ → ¬ θ))

Proof of Theorem msca

Step	Hyp	Ref	Expression
1		pm3.27 260	. . . 4 ⊢ ((φ ∧ ψ) → ψ)
2		msca.1	. . . . 5 ⊢ (φ → (ψ → χ))
3	2	imp 277	. . . 4 ⊢ ((φ ∧ ψ) → χ)
4	1, 3	jc 119	. . 3 ⊢ ((φ ∧ ψ) → ¬ (ψ → ¬ χ))
5		msca.2	. . 3 ⊢ (θ → (ψ → ¬ χ))
6	4, 5	nsyl 102	. 2 ⊢ ((φ ∧ ψ) → ¬ θ)
7	6	exp 291	1 ⊢ (φ → (ψ → ¬ θ))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196

This theorem is referenced by:  eqs1 828

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