Theorem bijadc 810

Description: Combine antecedents into a single biconditional. This inference is reminiscent of jadc 794. (Contributed by Jim Kingdon, 4-May-2018.)

Hypotheses

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

Assertion

Ref	Expression
bijadc	⊢ (DECID 𝜓 → ((𝜑 ↔ 𝜓) → 𝜒))

Proof of Theorem bijadc

Step	Hyp	Ref	Expression
1		bi2 128	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2		bijadc.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	syli 37	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜒))
4		bi1 116	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
5	4	con3d 594	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
6		bijadc.2	. . 3 ⊢ (¬ 𝜑 → (¬ 𝜓 → 𝜒))
7	5, 6	syli 37	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜓 → 𝜒))
8	3, 7	pm2.61ddc 792	1 ⊢ (DECID 𝜓 → ((𝜑 ↔ 𝜓) → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103  DECID wdc 776

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663

This theorem depends on definitions:  df-bi 115  df-dc 777

This theorem is referenced by: (None)