Theorem necon1ddc 2283

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

Hypothesis

Ref	Expression
necon1ddc.1	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)))

Assertion

Ref	Expression
necon1ddc	⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵)))

Proof of Theorem necon1ddc

Step	Hyp	Ref	Expression
1		df-ne 2206	. 2 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
2		necon1ddc.1	. . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)))
3	2	necon1bddc 2282	. 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐶 = 𝐷 → 𝐴 = 𝐵)))
4	1, 3	syl7bi 154	1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 ≠ 𝐷 → 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: (None)