Theorem necon4d 2953

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 2945	. 2 ⊢ (𝜑 → (𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵))
3		nne 2933	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
4	2, 3	imbitrdi 251	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ≠ wne 2929

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 207  df-ne 2930

This theorem is referenced by:  oa00  8480  map0g  8814  epfrs  9628  fin23lem24  10220  abs00  15198  oddvds  19461  01eq0ringOLD  20448  isdomn4  20633  isabvd  20729  uvcf1  21731  lindff1  21759  hausnei2  23269  dfconn2  23335  hausflimi  23896  hauspwpwf1  23903  cxpeq0  26615  his6  31081  fnpreimac  32655  deg1le0eq0  33543  lkreqN  39289  ltrnideq  40294  hdmapip0  42034  sticksstones2  42260  unitscyglem4  42311  rpnnen3  43149