Theorem frforeq3 4264

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 2201	. . . . . . 7 |- ( S = T -> ( y e. S <-> y e. T ) )
2	1	imbi2d 229	. . . . . 6 |- ( S = T -> ( ( y R x -> y e. S ) <-> ( y R x -> y e. T ) ) )
3	2	ralbidv 2435	. . . . 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 2201	. . . . 5 |- ( S = T -> ( x e. S <-> x e. T ) )
5	3, 4	imbi12d 233	. . . 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 2435	. . 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 3116	. . 3 |- ( S = T -> ( A C_ S <-> A C_ T ) )
8	6, 7	imbi12d 233	. 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 4248	. 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 4248	. 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 222	1 |- ( S = T -> (FrFor R A S <-> FrFor R A T ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2414   C_ wss 3066   class class class wbr 3924  FrFor wfrfor 4244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-in 3072  df-ss 3079  df-frfor 4248

This theorem is referenced by:  frind  4269