Theorem necon4d 2995

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 2987	. 2 ⊢ (𝜑 → (𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵))
3		nne 2975	. 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
4	2, 3	syl6ib 243	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1653   ≠ wne 2971

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 199  df-ne 2972

This theorem is referenced by:  oa00  7879  map0g  8136  epfrs  8857  fin23lem24  9432  abs00  14370  oddvds  18279  isabvd  19138  01eq0ring  19595  uvcf1  20456  lindff1  20484  hausnei2  21486  dfconn2  21551  hausflimi  22112  hauspwpwf1  22119  cxpeq0  24765  his6  28481  nn0sqeq1  30031  lkreqN  35191  ltrnideq  36196  hdmapip0  37936  rpnnen3  38384