Theorem frforeq2 4485

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 2749	. . . . 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 231	. . . 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 2761	. . 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 3271	. . 3 |- ( A = B -> ( A C_ T <-> B C_ T ) )
5	3, 4	imbi12d 234	. 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 4471	. 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 4471	. 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 223	1 |- ( A = B -> (FrFor R A T <-> FrFor R B T ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   class class class wbr 4125  FrFor wfrfor 4467

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-in 3226  df-ss 3233  df-frfor 4471

This theorem is referenced by:  freq2  4486