Theorem con4biddc 847

Description: A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.)

Hypothesis

Ref	Expression
con4biddc.1	⊢ (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜓 ↔ ¬ 𝜒))))

Assertion

Ref	Expression
con4biddc	⊢ (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜓 ↔ 𝜒))))

Proof of Theorem con4biddc

Step	Hyp	Ref	Expression
1		con4biddc.1	. . . . . 6 ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜓 ↔ ¬ 𝜒))))
2		biimpr 129	. . . . . 6 ⊢ ((¬ 𝜓 ↔ ¬ 𝜒) → (¬ 𝜒 → ¬ 𝜓))
3	1, 2	syl8 71	. . . . 5 ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜒 → ¬ 𝜓))))
4		condc 843	. . . . . 6 ⊢ (DECID 𝜒 → ((¬ 𝜒 → ¬ 𝜓) → (𝜓 → 𝜒)))
5	4	a2i 11	. . . . 5 ⊢ ((DECID 𝜒 → (¬ 𝜒 → ¬ 𝜓)) → (DECID 𝜒 → (𝜓 → 𝜒)))
6	3, 5	syl6 33	. . . 4 ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜓 → 𝜒))))
7	6	imp31 254	. . 3 ⊢ (((𝜑 ∧ DECID 𝜓) ∧ DECID 𝜒) → (𝜓 → 𝜒))
8		biimp 117	. . . . . 6 ⊢ ((¬ 𝜓 ↔ ¬ 𝜒) → (¬ 𝜓 → ¬ 𝜒))
9	1, 8	syl8 71	. . . . 5 ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜓 → ¬ 𝜒))))
10		condc 843	. . . . . 6 ⊢ (DECID 𝜓 → ((¬ 𝜓 → ¬ 𝜒) → (𝜒 → 𝜓)))
11	10	imim2d 54	. . . . 5 ⊢ (DECID 𝜓 → ((DECID 𝜒 → (¬ 𝜓 → ¬ 𝜒)) → (DECID 𝜒 → (𝜒 → 𝜓))))
12	9, 11	sylcom 28	. . . 4 ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜒 → 𝜓))))
13	12	imp31 254	. . 3 ⊢ (((𝜑 ∧ DECID 𝜓) ∧ DECID 𝜒) → (𝜒 → 𝜓))
14	7, 13	impbid 128	. 2 ⊢ (((𝜑 ∧ DECID 𝜓) ∧ DECID 𝜒) → (𝜓 ↔ 𝜒))
15	14	exp31 362	1 ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜓 ↔ 𝜒))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by:  necon4abiddc  2409