Theorem necon1ddc 2418

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 2341	. 2 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
2		necon1ddc.1	. . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)))
3	2	necon1bddc 2417	. 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐶 = 𝐷 → 𝐴 = 𝐵)))
4	1, 3	syl7bi 164	1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 829   = wceq 1348   ≠ wne 2340

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-ne 2341

This theorem is referenced by:  xblss2ps  13198  xblss2  13199  lgsne0  13733