Theorem frforeq1 4434

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

Assertion

Ref	Expression
frforeq1	|- ( R = S -> (FrFor R A T <-> FrFor S A T ) )

Proof of Theorem frforeq1

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

Step	Hyp	Ref	Expression
1		breq 4085	. . . . . . 7 |- ( R = S -> ( y R x <-> y S x ) )
2	1	imbi1d 231	. . . . . 6 |- ( R = S -> ( ( y R x -> y e. T ) <-> ( y S x -> y e. T ) ) )
3	2	ralbidv 2530	. . . . 5 |- ( R = S -> ( A. y e. A ( y R x -> y e. T ) <-> A. y e. A ( y S x -> y e. T ) ) )
4	3	imbi1d 231	. . . 4 |- ( R = S -> ( ( A. y e. A ( y R x -> y e. T ) -> x e. T ) <-> ( A. y e. A ( y S x -> y e. T ) -> x e. T ) ) )
5	4	ralbidv 2530	. . 3 |- ( R = S -> ( A. x e. A ( A. y e. A ( y R x -> y e. T ) -> x e. T ) <-> A. x e. A ( A. y e. A ( y S x -> y e. T ) -> x e. T ) ) )
6	5	imbi1d 231	. 2 |- ( R = S -> ( ( A. x e. A ( A. y e. A ( y R x -> y e. T ) -> x e. T ) -> A C_ T ) <-> ( A. x e. A ( A. y e. A ( y S x -> y e. T ) -> x e. T ) -> A C_ T ) ) )
7		df-frfor 4422	. 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 ) )
8		df-frfor 4422	. 2 |- (FrFor S A T <-> ( A. x e. A ( A. y e. A ( y S x -> y e. T ) -> x e. T ) -> A C_ T ) )
9	6, 7, 8	3bitr4g 223	1 |- ( R = S -> (FrFor R A T <-> FrFor S A T ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   class class class wbr 4083  FrFor wfrfor 4418

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-cleq 2222  df-clel 2225  df-ral 2513  df-br 4084  df-frfor 4422

This theorem is referenced by:  freq1  4435