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Theorem sbcexf 30758
Description: Move existential quantifier in and out of class substitution, with an explicit non-free variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
Hypothesis
Ref Expression
sbcexf.1  |-  F/_ y A
Assertion
Ref Expression
sbcexf  |-  ( [. A  /  x ]. E. y ph  <->  E. y [. A  /  x ]. ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)

Proof of Theorem sbcexf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1712 . . . 4  |-  F/ z
ph
21sb8e 2170 . . 3  |-  ( E. y ph  <->  E. z [ z  /  y ] ph )
32sbcbii 3380 . 2  |-  ( [. A  /  x ]. E. y ph  <->  [. A  /  x ]. E. z [ z  /  y ] ph )
4 sbcex2 3375 . 2  |-  ( [. A  /  x ]. E. z [ z  /  y ] ph  <->  E. z [. A  /  x ]. [ z  /  y ] ph )
5 sbcexf.1 . . . 4  |-  F/_ y A
6 nfs1v 2183 . . . 4  |-  F/ y [ z  /  y ] ph
75, 6nfsbc 3346 . . 3  |-  F/ y
[. A  /  x ]. [ z  /  y ] ph
8 nfv 1712 . . 3  |-  F/ z
[. A  /  x ]. ph
9 sbequ12r 1998 . . . 4  |-  ( z  =  y  ->  ( [ z  /  y ] ph  <->  ph ) )
109sbcbidv 3379 . . 3  |-  ( z  =  y  ->  ( [. A  /  x ]. [ z  /  y ] ph  <->  [. A  /  x ]. ph ) )
117, 8, 10cbvex 2027 . 2  |-  ( E. z [. A  /  x ]. [ z  / 
y ] ph  <->  E. y [. A  /  x ]. ph )
123, 4, 113bitri 271 1  |-  ( [. A  /  x ]. E. y ph  <->  E. y [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398   E.wex 1617   [wsb 1744   F/_wnfc 2602   [.wsbc 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-sbc 3325
This theorem is referenced by:  sbcexfi  30760
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