Theorem frforeq2 4363

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 2686	. . . . 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 2694	. . 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 3193	. . 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 4349	. 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 4349	. 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 1364   e. wcel 2160   A.wral 2468   C_ wss 3144   class class class wbr 4018  FrFor wfrfor 4345

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-in 3150  df-ss 3157  df-frfor 4349

This theorem is referenced by:  freq2  4364