Theorem necon1addc 2382

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 2307	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon1addc.1	. . 3 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 = 𝐵)))
3		con1dc 841	. . 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 164	1 ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 819   = wceq 1331   ≠ wne 2306

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-ne 2307

This theorem is referenced by: (None)