Theorem frforeq2 4323

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

Assertion

Ref	Expression
frforeq2	|- ( A = B -> (FrFor R A T <-> FrFor R B T ) )

Proof of Theorem frforeq2

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

Step	Hyp	Ref	Expression
1		raleq 2661	. . . . 5 |- ( A = B -> ( A. y e. A ( y R x -> y e. T ) <-> A. y e. B ( y R x -> y e. T ) ) )
2	1	imbi1d 230	. . . 4 |- ( A = B -> ( ( A. y e. A ( y R x -> y e. T ) -> x e. T ) <-> ( A. y e. B ( y R x -> y e. T ) -> x e. T ) ) )
3	2	raleqbi1dv 2669	. . 3 |- ( A = B -> ( A. x e. A ( A. y e. A ( y R x -> y e. T ) -> x e. T ) <-> A. x e. B ( A. y e. B ( y R x -> y e. T ) -> x e. T ) ) )
4		sseq1 3165	. . 3 |- ( A = B -> ( A C_ T <-> B C_ T ) )
5	3, 4	imbi12d 233	. 2 |- ( A = B -> ( ( A. x e. A ( A. y e. A ( y R x -> y e. T ) -> x e. T ) -> A C_ T ) <-> ( A. x e. B ( A. y e. B ( y R x -> y e. T ) -> x e. T ) -> B C_ T ) ) )
6		df-frfor 4309	. 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 ) )
7		df-frfor 4309	. 2 |- (FrFor R B T <-> ( A. x e. B ( A. y e. B ( y R x -> y e. T ) -> x e. T ) -> B C_ T ) )
8	5, 6, 7	3bitr4g 222	1 |- ( A = B -> (FrFor R A T <-> FrFor R B T ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   C_ wss 3116   class class class wbr 3982  FrFor wfrfor 4305

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-in 3122  df-ss 3129  df-frfor 4309

This theorem is referenced by:  freq2  4324