Theorem necon4d 2959

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 2951	. 2 ⊢ (𝜑 → (𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵))
3		nne 2939	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
4	2, 3	imbitrdi 252	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1547   ≠ wne 2935

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 208  df-ne 2936

This theorem is referenced by:  oa00  8491  map0g  8829  epfrs  9650  fin23lem24  10242  abs00  15249  oddvds  19520  01eq0ringOLD  20510  isdomn4  20695  isabvd  20791  uvcf1  21774  lindff1  21802  hausnei2  23343  dfconn2  23409  hausflimi  23970  hauspwpwf1  23977  cxpeq0  26667  his6  31195  fnpreimac  32769  deg1le0eq0  33663  lkreqN  39669  ltrnideq  40674  hdmapip0  42414  sticksstones2  42639  unitscyglem4  42690  rpnnen3  43484