Theorem necon3abid 1602

Description: Deduction from equality to inequality.

Hypothesis

Ref	Expression
necon3abid.1	|- (ph -> (A = B <-> ps))

Assertion

Ref	Expression
necon3abid	|- (ph -> (A =/= B <-> -. ps))

Proof of Theorem necon3abid

Step	Hyp	Ref	Expression
1		necon3abid.1	. . 3 |- (ph -> (A = B <-> ps))
2	1	negbid 613	. 2 |- (ph -> (-. A = B <-> -. ps))
3		df-ne 1590	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	syl5bb 534	1 |- (ph -> (A =/= B <-> -. ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 958   =/= wne 1588

This theorem is referenced by:  necon3bbid 1603  necon3bid 1604  foconst 3689  om00el 4213  cardsdom 4847  nmlno0lem 8449  nmlnop0ALT 9915  atcvat2 10309

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 147  df-an 225  df-ne 1590

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