Theorem frforeq2 4393

Description: Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)

Assertion

Ref	Expression
frforeq2	|- ( A = B -> (FrFor R A T <-> FrFor R B T ) )

Proof of Theorem frforeq2

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

Step	Hyp	Ref	Expression
1		raleq 2702	. . . . 5 |- ( A = B -> ( A. y e. A ( y R x -> y e. T ) <-> A. y e. B ( y R x -> y e. T ) ) )
2	1	imbi1d 231	. . . 4 |- ( A = B -> ( ( A. y e. A ( y R x -> y e. T ) -> x e. T ) <-> ( A. y e. B ( y R x -> y e. T ) -> x e. T ) ) )
3	2	raleqbi1dv 2714	. . 3 |- ( A = B -> ( A. x e. A ( A. y e. A ( y R x -> y e. T ) -> x e. T ) <-> A. x e. B ( A. y e. B ( y R x -> y e. T ) -> x e. T ) ) )
4		sseq1 3216	. . 3 |- ( A = B -> ( A C_ T <-> B C_ T ) )
5	3, 4	imbi12d 234	. 2 |- ( A = B -> ( ( A. x e. A ( A. y e. A ( y R x -> y e. T ) -> x e. T ) -> A C_ T ) <-> ( A. x e. B ( A. y e. B ( y R x -> y e. T ) -> x e. T ) -> B C_ T ) ) )
6		df-frfor 4379	. 2 |- (FrFor R A T <-> ( A. x e. A ( A. y e. A ( y R x -> y e. T ) -> x e. T ) -> A C_ T ) )
7		df-frfor 4379	. 2 |- (FrFor R B T <-> ( A. x e. B ( A. y e. B ( y R x -> y e. T ) -> x e. T ) -> B C_ T ) )
8	5, 6, 7	3bitr4g 223	1 |- ( A = B -> (FrFor R A T <-> FrFor R B T ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2176   A.wral 2484   C_ wss 3166   class class class wbr 4045  FrFor wfrfor 4375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-in 3172  df-ss 3179  df-frfor 4379

This theorem is referenced by:  freq2  4394