Theorem necon3bi 2045

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

Hypothesis

Ref	Expression
necon3bi.1	|- (A = B -> ph)

Assertion

Ref	Expression
necon3bi	|- (-. ph -> A =/= B)

Proof of Theorem necon3bi

Step	Hyp	Ref	Expression
1		nne 2021	. . 3 |- (-. A =/= B <-> A = B)
2		necon3bi.1	. . 3 |- (A = B -> ph)
3	1, 2	sylbi 216	. 2 |- (-. A =/= B -> ph)
4	3	con1i 112	1 |- (-. ph -> A =/= B)

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

This theorem is referenced by:  r19.2zb 2959  alephord 6023  acdc3lem 8754  acdc2lem1 8757  acdclem 8763  alexsublem2 15438  alexsublem4 15440  fimax 15746  heiborlem21 15975

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