Theorem necon1addc 2456

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

Hypothesis

Ref	Expression
necon1addc.1	⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 = 𝐵)))

Assertion

Ref	Expression
necon1addc	⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 → 𝜓)))

Proof of Theorem necon1addc

Step	Hyp	Ref	Expression
1		df-ne 2381	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon1addc.1	. . 3 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 = 𝐵)))
3		con1dc 860	. . 3 ⊢ (DECID 𝜓 → ((¬ 𝜓 → 𝐴 = 𝐵) → (¬ 𝐴 = 𝐵 → 𝜓)))
4	2, 3	sylcom 28	. 2 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝐴 = 𝐵 → 𝜓)))
5	1, 4	syl7bi 165	1 ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 838   = wceq 1375   ≠ wne 2380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713

This theorem depends on definitions:  df-bi 117  df-stab 835  df-dc 839  df-ne 2381

This theorem is referenced by:  seqf1oglem1  10708  seqf1oglem2  10709