Theorem necon4bddc 2407

Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)

Hypothesis

Ref	Expression
necon4bddc.1	⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 ≠ 𝐵)))

Assertion

Ref	Expression
necon4bddc	⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 → 𝜓)))

Proof of Theorem necon4bddc

Step	Hyp	Ref	Expression
1		necon4bddc.1	. . 3 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 ≠ 𝐵)))
2		df-ne 2337	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	syl8ib 165	. 2 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → ¬ 𝐴 = 𝐵)))
4		condc 843	. 2 ⊢ (DECID 𝜓 → ((¬ 𝜓 → ¬ 𝐴 = 𝐵) → (𝐴 = 𝐵 → 𝜓)))
5	3, 4	sylcom 28	1 ⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 824   = wceq 1343   ≠ wne 2336

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-ne 2337

This theorem is referenced by: (None)