Theorem necon1bi 2048

Description: Contrapositive inference for inequality. (The proof was shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon1bi.1	|- (A =/= B -> ph)

Assertion

Ref	Expression
necon1bi	|- (-. ph -> A = B)

Proof of Theorem necon1bi

Step	Hyp	Ref	Expression
1		necon1bi.1	. . 3 |- (A =/= B -> ph)
2	1	con3i 114	. 2 |- (-. ph -> -. A =/= B)
3		nne 2021	. 2 |- (-. A =/= B <-> A = B)
4	2, 3	sylib 215	1 |- (-. ph -> A = B)

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   =/= wne 2017

This theorem is referenced by:  unissintOLD 3242  peano5 3975  1st2val 5038  2nd2val 5039  eceqopreq 5372  mapprc 5385  pw2en 5505  setind 5759  isumnul 8464  hatomistici 11934

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-ne 2019

Copyright terms: Public domain