Theorem sfineq1 4527

Description: Equality theorem for the finite S relationship. (Contributed by SF, 27-Jan-2015.)

Assertion

Ref	Expression
sfineq1	|- (A = B -> ( Sfin (A, C) <-> Sfin (B, C)))

Proof of Theorem sfineq1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2413	. . 3 |- (A = B -> (A e. Nn <-> B e. Nn ))
2		eleq2 2414	. . . . 5 |- (A = B -> (~P1 y e. A <-> ~P1 y e. B))
3	2	anbi1d 685	. . . 4 |- (A = B -> ((~P1 y e. A /\ ~Py e. C) <-> (~P1 y e. B /\ ~Py e. C)))
4	3	exbidv 1626	. . 3 |- (A = B -> (E.y(~P1 y e. A /\ ~Py e. C) <-> E.y(~P1 y e. B /\ ~Py e. C)))
5	1, 4	3anbi13d 1254	. 2 |- (A = B -> ((A e. Nn /\ C e. Nn /\ E.y(~P1 y e. A /\ ~Py e. C)) <-> (B e. Nn /\ C e. Nn /\ E.y(~P1 y e. B /\ ~Py e. C))))
6		df-sfin 4447	. 2 |- ( Sfin (A, C) <-> (A e. Nn /\ C e. Nn /\ E.y(~P1 y e. A /\ ~Py e. C)))
7		df-sfin 4447	. 2 |- ( Sfin (B, C) <-> (B e. Nn /\ C e. Nn /\ E.y(~P1 y e. B /\ ~Py e. C)))
8	5, 6, 7	3bitr4g 279	1 |- (A = B -> ( Sfin (A, C) <-> Sfin (B, C)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  ~Pcpw 3723  ~P₁ cpw1 4136   Nn cnnc 4374   S_fin wsfin 4439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-cleq 2346  df-clel 2349  df-sfin 4447

This theorem is referenced by:  sfintfinlem1  4532  sfintfin  4533  spfinsfincl  4540  vfinspsslem1  4551