Theorem r19.12sn 3660

Description: Special case of r19.12 2583 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 3043	. 2 |- ( A e. V -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
2		rexsns 3633	. 2 |- ( E. x e. { A } A. y e. B ph <-> [. A / x ]. A. y e. B ph )
3		rexsns 3633	. . 3 |- ( E. x e. { A } ph <-> [. A / x ]. ph )
4	3	ralbii 2483	. 2 |- ( A. y e. B E. x e. { A } ph <-> A. y e. B [. A / x ]. ph )
5	1, 2, 4	3bitr4g 223	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 105   e. wcel 2148   A.wral 2455   E.wrex 2456   [.wsbc 2964   {csn 3594

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-sn 3600

This theorem is referenced by: (None)