Theorem jadc 748

Description: Inference forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 25-Mar-2018.)

Hypotheses

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

Assertion

Ref	Expression
jadc	⊢ (DECID φ → ((φ → ψ) → χ))

Proof of Theorem jadc

Step	Hyp	Ref	Expression
1		jadc.2	. . 3 ⊢ (ψ → χ)
2	1	imim2i 12	. 2 ⊢ ((φ → ψ) → (φ → χ))
3		jadc.1	. . 3 ⊢ (DECID φ → (¬ φ → χ))
4		pm2.6dc 747	. . 3 ⊢ (DECID φ → ((¬ φ → χ) → ((φ → χ) → χ)))
5	3, 4	mpd 13	. 2 ⊢ (DECID φ → ((φ → χ) → χ))
6	2, 5	syl5 28	1 ⊢ (DECID φ → ((φ → ψ) → χ))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 730

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 617

This theorem depends on definitions:  df-bi 110  df-dc 731

This theorem is referenced by:  pm5.71dc  854