Theorem frforeq3 4084

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

Assertion

Ref	Expression
frforeq3	|- ( S = T -> (FrFor R A S <-> FrFor R A T ) )

Proof of Theorem frforeq3

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

Step	Hyp	Ref	Expression
1		eleq2 2101	. . . . . . 7 |- ( S = T -> ( y e. S <-> y e. T ) )
2	1	imbi2d 219	. . . . . 6 |- ( S = T -> ( ( y R x -> y e. S ) <-> ( y R x -> y e. T ) ) )
3	2	ralbidv 2326	. . . . 5 |- ( S = T -> ( A. y e. A ( y R x -> y e. S ) <-> A. y e. A ( y R x -> y e. T ) ) )
4		eleq2 2101	. . . . 5 |- ( S = T -> ( x e. S <-> x e. T ) )
5	3, 4	imbi12d 223	. . . 4 |- ( S = T -> ( ( A. y e. A ( y R x -> y e. S ) -> x e. S ) <-> ( A. y e. A ( y R x -> y e. T ) -> x e. T ) ) )
6	5	ralbidv 2326	. . 3 |- ( S = T -> ( A. x e. A ( A. y e. A ( y R x -> y e. S ) -> x e. S ) <-> A. x e. A ( A. y e. A ( y R x -> y e. T ) -> x e. T ) ) )
7		sseq2 2967	. . 3 |- ( S = T -> ( A C_ S <-> A C_ T ) )
8	6, 7	imbi12d 223	. 2 |- ( S = T -> ( ( A. x e. A ( A. y e. A ( y R x -> y e. S ) -> x e. S ) -> A C_ S ) <-> ( A. x e. A ( A. y e. A ( y R x -> y e. T ) -> x e. T ) -> A C_ T ) ) )
9		df-frfor 4068	. 2 |- (FrFor R A S <-> ( A. x e. A ( A. y e. A ( y R x -> y e. S ) -> x e. S ) -> A C_ S ) )
10		df-frfor 4068	. 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 ) )
11	8, 9, 10	3bitr4g 212	1 |- ( S = T -> (FrFor R A S <-> FrFor R A T ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   C_ wss 2917   class class class wbr 3764  FrFor wfrfor 4064

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-in 2924  df-ss 2931  df-frfor 4068

This theorem is referenced by:  frind  4089