Theorem necon4aidc 2408

Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)

Hypothesis

Ref	Expression
necon4aidc.1	⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜑))

Assertion

Ref	Expression
necon4aidc	⊢ (DECID 𝐴 = 𝐵 → (𝜑 → 𝐴 = 𝐵))

Proof of Theorem necon4aidc

Step	Hyp	Ref	Expression
1		df-ne 2341	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon4aidc.1	. . 3 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜑))
3	1, 2	syl5bir 152	. 2 ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → ¬ 𝜑))
4		condc 848	. 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 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:  necon4idc  2409