Theorem necon2d 2955

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 2933	. . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
3	1, 2	imbitrdi 251	. 2 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷))
4	3	necon2ad 2947	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ≠ wne 2932

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 207  df-ne 2933

This theorem is referenced by:  map0g  8832  cantnf  9614  hashprg  14357  bcthlem5  25295  deg1ldgn  26058  cxpeq0  26642  lfgrn1cycl  29873  uspgrn2crct  29876  poimirlem17  37958  poimirlem20  37961  poimirlem22  37963  poimirlem27  37968  islshpat  39463  cdleme18b  40738  cdlemh  41263  prjspner1  43059