Theorem freq1 5084

Description: Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)

Assertion

Ref	Expression
freq1	|- ( R = S -> ( R Fr A <-> S Fr A ) )

Proof of Theorem freq1

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 4655	. . . . . 6 |- ( R = S -> ( z R y <-> z S y ) )
2	1	notbid 308	. . . . 5 |- ( R = S -> ( -. z R y <-> -. z S y ) )
3	2	rexralbidv 3058	. . . 4 |- ( R = S -> ( E. y e. x A. z e. x -. z R y <-> E. y e. x A. z e. x -. z S y ) )
4	3	imbi2d 330	. . 3 |- ( R = S -> ( ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) <-> ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. z S y ) ) )
5	4	albidv 1849	. 2 |- ( R = S -> ( A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. z S y ) ) )
6		df-fr 5073	. 2 |- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) )
7		df-fr 5073	. 2 |- ( S Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x A. z e. x -. z S y ) )
8	5, 6, 7	3bitr4g 303	1 |- ( R = S -> ( R Fr A <-> S Fr A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Fr wfr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-ral 2917  df-rex 2918  df-br 4654  df-fr 5073

This theorem is referenced by:  weeq1  5102  freq12d  37609