Theorem necon4d 2957

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 2949	. 2 ⊢ (𝜑 → (𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵))
3		nne 2937	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
4	2, 3	imbitrdi 251	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ≠ wne 2933

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 2934

This theorem is referenced by:  oa00  8496  map0g  8834  epfrs  9652  fin23lem24  10244  abs00  15224  oddvds  19488  01eq0ringOLD  20476  isdomn4  20661  isabvd  20757  uvcf1  21759  lindff1  21787  hausnei2  23309  dfconn2  23375  hausflimi  23936  hauspwpwf1  23943  cxpeq0  26655  his6  31186  fnpreimac  32759  deg1le0eq0  33665  lkreqN  39540  ltrnideq  40545  hdmapip0  42285  sticksstones2  42511  unitscyglem4  42562  rpnnen3  43383