Theorem nebidc 2420

Description: Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)

Assertion

Ref	Expression
nebidc	⊢ (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))))

Proof of Theorem nebidc

Step	Hyp	Ref	Expression
1		id 19	. . . 4 ⊢ ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))
2	1	necon3bid 2381	. . 3 ⊢ ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
3		id 19	. . . . . . . 8 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
4	3	a1d 22	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)))
5	4	a1d 22	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))))
6	5	necon4biddc 2415	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))))
7	6	com3l 81	. . . 4 ⊢ (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → ((𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))))
8	7	imp 123	. . 3 ⊢ ((DECID 𝐴 = 𝐵 ∧ DECID 𝐶 = 𝐷) → ((𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)))
9	2, 8	impbid2 142	. 2 ⊢ ((DECID 𝐴 = 𝐵 ∧ DECID 𝐶 = 𝐷) → ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)))
10	9	ex 114	1 ⊢ (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 829   = wceq 1348   ≠ wne 2340

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-ne 2341

This theorem is referenced by:  rpexp  12107