Theorem necon1bddc 2282

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

Hypothesis

Ref	Expression
necon1bddc.1	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜓)))

Assertion

Ref	Expression
necon1bddc	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 → 𝐴 = 𝐵)))

Proof of Theorem necon1bddc

Step	Hyp	Ref	Expression
1		necon1bddc.1	. . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜓)))
2		df-ne 2206	. . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	2	imbi1i 227	. . 3 ⊢ ((𝐴 ≠ 𝐵 → 𝜓) ↔ (¬ 𝐴 = 𝐵 → 𝜓))
4	1, 3	syl6ib 150	. 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → 𝜓)))
5		con1dc 753	. 2 ⊢ (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵 → 𝜓) → (¬ 𝜓 → 𝐴 = 𝐵)))
6	4, 5	sylcom 25	1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 → 𝐴 = 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 742   = wceq 1243   ≠ wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743  df-ne 2206

This theorem is referenced by:  necon1ddc  2283