Theorem frforeq3 4349

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 2241	. . . . . . 7 |- ( S = T -> ( y e. S <-> y e. T ) )
2	1	imbi2d 230	. . . . . 6 |- ( S = T -> ( ( y R x -> y e. S ) <-> ( y R x -> y e. T ) ) )
3	2	ralbidv 2477	. . . . 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 2241	. . . . 5 |- ( S = T -> ( x e. S <-> x e. T ) )
5	3, 4	imbi12d 234	. . . 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 2477	. . 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 3181	. . 3 |- ( S = T -> ( A C_ S <-> A C_ T ) )
8	6, 7	imbi12d 234	. 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 4333	. 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 4333	. 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 223	1 |- ( S = T -> (FrFor R A S <-> FrFor R A T ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   C_ wss 3131   class class class wbr 4005  FrFor wfrfor 4329

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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-in 3137  df-ss 3144  df-frfor 4333

This theorem is referenced by:  frind  4354