Theorem necon2d 2805

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 2782	. . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
3	1, 2	syl6ib 240	. 2 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷))
4	3	necon2ad 2797	1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ≠ wne 2780

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 196  df-ne 2782

This theorem is referenced by:  map0g  7783  cantnf  8473  hashprg  13043  hashprgOLD  13044  bcthlem5  22933  deg1ldgn  23657  cxpeq0  24224  poimirlem17  32596  poimirlem20  32599  poimirlem22  32601  poimirlem27  32606  islshpat  33322  cdleme18b  34597  cdlemh  35123  lfgrn1cycl  41008  uspgrn2crct  41011