Theorem necon4d 2984

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 2976	. 2 ⊢ (𝜑 → (𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵))
3		nne 2964	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
4	2, 3	imbitrdi 254	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ≠ wne 2960

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-ne 2961

This theorem is used by:  oa00  8546  map0g  8884  epfrs  9703  fin23lem24  10317  abs00  15359  oddvds  19640  01eq0ringOLD  20658  isdomn4  20843  isabvd  20944  uvcf1  21971  lindff1  21999  hausnei2  23539  dfconn2  23605  hausflimi  24166  hauspwpwf1  24173  cxpeq0  26872  his6  31480  fnpreimac  33044  deg1le0eq0  33886  lkreqN  39977  ltrnideq  40982  hdmapip0  42722  sticksstones2  42947  unitscyglem4  42998  rpnnen3  43792