Theorem r19.12sn 3649

Description: Special case of r19.12 2576 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 20-Dec-2021.)

Assertion

Ref	Expression
r19.12sn	|- ( A e. V -> ( E. x e. { A } A. y e. B ph <-> A. y e. B E. x e. { A } ph ) )

Distinct variable groups:   x, y, A   x, B

Allowed substitution hints:   ph( x, y)   B( y)   V( x, y)

Proof of Theorem r19.12sn

Step	Hyp	Ref	Expression
1		sbcralg 3033	. 2 |- ( A e. V -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
2		rexsns 3622	. 2 |- ( E. x e. { A } A. y e. B ph <-> [. A / x ]. A. y e. B ph )
3		rexsns 3622	. . 3 |- ( E. x e. { A } ph <-> [. A / x ]. ph )
4	3	ralbii 2476	. 2 |- ( A. y e. B E. x e. { A } ph <-> A. y e. B [. A / x ]. ph )
5	1, 2, 4	3bitr4g 222	1 |- ( A e. V -> ( E. x e. { A } A. y e. B ph <-> A. y e. B E. x e. { A } ph ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2141   A.wral 2448   E.wrex 2449   [.wsbc 2955   {csn 3583

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-sn 3589

This theorem is referenced by: (None)