Theorem necon2d 2981

Description: Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)

Hypothesis

Ref	Expression
necon2d.1	⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 ≠ 𝐷))

Assertion

Ref	Expression
necon2d	⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵))

Proof of Theorem necon2d

Step	Hyp	Ref	Expression
1		necon2d.1	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 ≠ 𝐷))
2		df-ne 2959	. . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
3	1, 2	imbitrdi 254	. 2 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷))
4	3	necon2ad 2973	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ≠ wne 2958

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 210  df-ne 2959

This theorem is referenced by:  map0g  8883  cantnf  9663  hashprg  14433  bcthlem5  25468  deg1ldgn  26231  cxpeq0  26821  lfgrn1cycl  30132  uspgrn2crct  30135  poimirlem17  38266  poimirlem20  38269  poimirlem22  38271  poimirlem27  38276  islshpat  39769  cdleme18b  41044  cdlemh  41569  prjspner1  43338