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Theorem necon4ai 2705
Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
Hypothesis
Ref Expression
necon4ai.1  |-  ( A  =/=  B  ->  -.  ph )
Assertion
Ref Expression
necon4ai  |-  ( ph  ->  A  =  B )

Proof of Theorem necon4ai
StepHypRef Expression
1 notnot1 122 . 2  |-  ( ph  ->  -.  -.  ph )
2 necon4ai.1 . . 3  |-  ( A  =/=  B  ->  -.  ph )
32necon1bi 2700 . 2  |-  ( -. 
-.  ph  ->  A  =  B )
41, 3syl 16 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379    =/= wne 2662
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 185  df-ne 2664
This theorem is referenced by:  necon4i  2711  dmsn0el  5483  cfeq0  8648
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