Theorem jaddc 760

Description: Deduction forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 26-Mar-2018.)

Hypotheses

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

Assertion

Ref	Expression
jaddc	⊢ (φ → (DECID ψ → ((ψ → χ) → θ)))

Proof of Theorem jaddc

Step	Hyp	Ref	Expression
1		jaddc.2	. . 3 ⊢ (φ → (χ → θ))
2	1	imim2d 48	. 2 ⊢ (φ → ((ψ → χ) → (ψ → θ)))
3		jaddc.1	. . 3 ⊢ (φ → (DECID ψ → (¬ ψ → θ)))
4		pm2.6dc 758	. . 3 ⊢ (DECID ψ → ((¬ ψ → θ) → ((ψ → θ) → θ)))
5	3, 4	sylcom 25	. 2 ⊢ (φ → (DECID ψ → ((ψ → θ) → θ)))
6	2, 5	syl5d 62	1 ⊢ (φ → (DECID ψ → ((ψ → χ) → θ)))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 741

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by:  pm2.54dc  789