Theorem necon4ai 2825

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

Step	Hyp	Ref	Expression
1		notnot 136	. 2 |- ( ph -> -. -. ph )
2		necon4ai.1	. . 3 |- ( A =/= B -> -. ph )
3	2	necon1bi 2822	. 2 |- ( -. -. ph -> A = B )
4	1, 3	syl 17	1 |- ( ph -> A = B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   =/= wne 2794

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 197  df-ne 2795

This theorem is referenced by:  necon4i  2829  dmsn0el  5604  funsneqopb  6419  cfeq0  9078