Theorem r19.12sn 3789

Description: Special case of r19.12 2727 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypothesis

Ref	Expression
r19.12sn.1	|- A e. _V

Assertion

Ref	Expression
r19.12sn	|- (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)

Proof of Theorem r19.12sn

Step	Hyp	Ref	Expression
1		r19.12sn.1	. 2 |- A e. _V
2		sbcralg 3120	. . 3 |- (A e. _V -> ( [.A / x ].A.y e. B ph <-> A.y e. B [.A / x ].ph))
3		rexsns 3764	. . 3 |- (A e. _V -> (E.x e. {A}A.y e. B ph <-> [.A / x ].A.y e. B ph))
4		rexsns 3764	. . . 4 |- (A e. _V -> (E.x e. {A}ph <-> [.A / x ].ph))
5	4	ralbidv 2634	. . 3 |- (A e. _V -> (A.y e. B E.x e. {A}ph <-> A.y e. B [.A / x ].ph))
6	2, 3, 5	3bitr4d 276	. 2 |- (A e. _V -> (E.x e. {A}A.y e. B ph <-> A.y e. B E.x e. {A}ph))
7	1, 6	ax-mp 5	1 |- (E.x e. {A}A.y e. B ph <-> A.y e. B E.x e. {A}ph)

Syntax hints:   <-> wb 176   e. wcel 1710  A.wral 2614  E.wrex 2615  _Vcvv 2859   [.wsbc 3046  {csn 3737

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-sn 3741

This theorem is referenced by: (None)