Theorem necon2ai 2562

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon2ai.1	⊢ (A = B → ¬ φ)

Assertion

Ref	Expression
necon2ai	⊢ (φ → A ≠ B)

Proof of Theorem necon2ai

Step	Hyp	Ref	Expression
1		nne 2521	. . 3 ⊢ (¬ A ≠ B ↔ A = B)
2		necon2ai.1	. . 3 ⊢ (A = B → ¬ φ)
3	1, 2	sylbi 187	. 2 ⊢ (¬ A ≠ B → ¬ φ)
4	3	con4i 122	1 ⊢ (φ → A ≠ B)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ≠ wne 2517

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 177  df-ne 2519

This theorem is referenced by:  necon2i  2564  neneqad  2587  peano4  4558  addccan2  4560  ncssfin  6152  ncspw1eu  6160  nntccl  6171  ceclb  6184  ce0ncpw1  6186  cecl  6187  nclecid  6198  le0nc  6201  addlecncs  6210