Theorem necon4d 2960

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 2952	. 2 ⊢ (𝜑 → (𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵))
3		nne 2940	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
4	2, 3	imbitrdi 250	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1534   ≠ wne 2936

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 2937

This theorem is referenced by:  oa00  8574  map0g  8897  epfrs  9749  fin23lem24  10340  abs00  15263  oddvds  19496  01eq0ringOLD  20462  isabvd  20694  isdomn4  21244  uvcf1  21720  lindff1  21748  hausnei2  23251  dfconn2  23317  hausflimi  23878  hauspwpwf1  23885  cxpeq0  26606  his6  30903  fnpreimac  32451  deg1le0eq0  33248  lkreqN  38637  ltrnideq  39643  hdmapip0  41383  sticksstones2  41614  rpnnen3  42444