Theorem necon2d 2969

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 2947	. . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
3	1, 2	imbitrdi 251	. 2 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷))
4	3	necon2ad 2961	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ≠ wne 2946

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 2947

This theorem is referenced by:  map0g  8942  cantnf  9762  hashprg  14444  bcthlem5  25381  deg1ldgn  26152  cxpeq0  26738  lfgrn1cycl  29838  uspgrn2crct  29841  poimirlem17  37597  poimirlem20  37600  poimirlem22  37602  poimirlem27  37607  islshpat  38973  cdleme18b  40249  cdlemh  40774  prjspner1  42581