Theorem necon1bidc 2297

Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)

Hypothesis

Ref	Expression
necon1bidc.1	⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑))

Assertion

Ref	Expression
necon1bidc	⊢ (DECID 𝐴 = 𝐵 → (¬ 𝜑 → 𝐴 = 𝐵))

Proof of Theorem necon1bidc

Step	Hyp	Ref	Expression
1		df-ne 2246	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon1bidc.1	. . 3 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑))
3	1, 2	syl5bir 151	. 2 ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → 𝜑))
4		con1dc 786	. 2 ⊢ (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵 → 𝜑) → (¬ 𝜑 → 𝐴 = 𝐵)))
5	3, 4	mpd 13	1 ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝜑 → 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 775   = wceq 1284   ≠ wne 2245

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 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776  df-ne 2246

This theorem is referenced by: (None)