Theorem r19.12sn 3748

Description: Special case of r19.12 2649 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 3120	. 2 |- ( A e. V -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
2		rexsns 3721	. 2 |- ( E. x e. { A } A. y e. B ph <-> [. A / x ]. A. y e. B ph )
3		rexsns 3721	. . 3 |- ( E. x e. { A } ph <-> [. A / x ]. ph )
4	3	ralbii 2548	. 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 2203   A.wral 2520   E.wrex 2521   [.wsbc 3041   {csn 3682

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-sn 3688

This theorem is referenced by: (None)