Theorem necon4d 2949

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 2941	. 2 ⊢ (𝜑 → (𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵))
3		nne 2929	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
4	2, 3	imbitrdi 251	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ≠ wne 2925

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 2926

This theorem is referenced by:  oa00  8484  map0g  8818  epfrs  9646  fin23lem24  10235  abs00  15214  oddvds  19444  01eq0ringOLD  20434  isdomn4  20619  isabvd  20715  uvcf1  21717  lindff1  21745  hausnei2  23256  dfconn2  23322  hausflimi  23883  hauspwpwf1  23890  cxpeq0  26603  his6  31061  fnpreimac  32628  deg1le0eq0  33518  lkreqN  39148  ltrnideq  40154  hdmapip0  41894  sticksstones2  42120  unitscyglem4  42171  rpnnen3  43005