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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 V
Assertion
Ref Expression
r19.12sn (x {A}y B φy B x {A}φ)
Distinct variable groups:   x,y,A   x,B
Allowed substitution hints:   φ(x,y)   B(y)

Proof of Theorem r19.12sn
StepHypRef Expression
1 r19.12sn.1 . 2 A V
2 sbcralg 3120 . . 3 (A V → ([̣A / xy B φy BA / xφ))
3 rexsns 3764 . . 3 (A V → (x {A}y B φ ↔ [̣A / xy B φ))
4 rexsns 3764 . . . 4 (A V → (x {A}φ ↔ [̣A / xφ))
54ralbidv 2634 . . 3 (A V → (y B x {A}φy BA / xφ))
62, 3, 53bitr4d 276 . 2 (A V → (x {A}y B φy B x {A}φ))
71, 6ax-mp 8 1 (x {A}y B φy B x {A}φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  Vcvv 2859  [̣wsbc 3046  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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)
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