Theorem frforeq1 4446

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 4095	. . . . . . 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 2533	. . . . 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 2533	. . 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 4434	. 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 4434	. 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 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   class class class wbr 4093  FrFor wfrfor 4430

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-cleq 2224  df-clel 2227  df-ral 2516  df-br 4094  df-frfor 4434

This theorem is referenced by:  freq1  4447