Theorem necon4d 2966

Description: Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon4d.1	⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷))

Assertion

Ref	Expression
necon4d	⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵))

Proof of Theorem necon4d

Step	Hyp	Ref	Expression
1		necon4d.1	. . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷))
2	1	necon2bd 2958	. 2 ⊢ (𝜑 → (𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵))
3		nne 2946	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
4	2, 3	syl6ib 250	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ≠ wne 2942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-ne 2943

This theorem is referenced by:  oa00  8352  map0g  8630  epfrs  9420  fin23lem24  10009  abs00  14929  oddvds  19070  isabvd  19995  01eq0ring  20456  uvcf1  20909  lindff1  20937  hausnei2  22412  dfconn2  22478  hausflimi  23039  hauspwpwf1  23046  cxpeq0  25738  his6  29362  fnpreimac  30910  lkreqN  37111  ltrnideq  38116  hdmapip0  39856  sticksstones2  40031  isdomn4  40100  rpnnen3  40770