Theorem necon4d 2979

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 2971	. 2 ⊢ (𝜑 → (𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵))
3		nne 2959	. 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 2955

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 2956

This theorem is used by:  oa00  8546  map0g  8891  epfrs  9710  fin23lem24  10324  abs00  15376  oddvds  19674  01eq0ringOLD  20692  isdomn4  20877  isabvd  20978  uvcf1  22005  lindff1  22033  hausnei2  23578  dfconn2  23644  hausflimi  24206  hauspwpwf1  24213  cxpeq0  26915  his6  31580  fnpreimac  33143  deg1le0eq0  33983  lkreqN  40043  ltrnideq  41048  hdmapip0  42788  sticksstones2  43013  unitscyglem4  43064  rpnnen3  43873