Theorem r19.12sn 3555

Description: Special case of r19.12 2512 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 2955	. 2 |- ( A e. V -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
2		rexsns 3529	. 2 |- ( E. x e. { A } A. y e. B ph <-> [. A / x ]. A. y e. B ph )
3		rexsns 3529	. . 3 |- ( E. x e. { A } ph <-> [. A / x ]. ph )
4	3	ralbii 2415	. 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 1463   A.wral 2390   E.wrex 2391   [.wsbc 2878   {csn 3493

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-sn 3499

This theorem is referenced by: (None)