Theorem necon2d 2952

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 2930	. . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
3	1, 2	imbitrdi 251	. 2 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷))
4	3	necon2ad 2944	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ≠ wne 2929

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 2930

This theorem is referenced by:  map0g  8814  cantnf  9590  hashprg  14304  bcthlem5  25256  deg1ldgn  26026  cxpeq0  26615  lfgrn1cycl  29785  uspgrn2crct  29788  poimirlem17  37697  poimirlem20  37700  poimirlem22  37702  poimirlem27  37707  islshpat  39136  cdleme18b  40411  cdlemh  40936  prjspner1  42744