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Theorem sbcexf 30108
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 sbcexf.1 . . . 4  |-  F/_ y A
2 nfe1 1784 . . . 4  |-  F/ y E. y ph
31, 2nfsbc 3346 . . 3  |-  F/ y
[. A  /  x ]. E. y ph
4 nfe1 1784 . . 3  |-  F/ y E. y [. A  /  x ]. ph
53, 4nfbi 1876 . 2  |-  F/ y ( [. A  /  x ]. E. y ph  <->  E. y [. A  /  x ]. ph )
6 nfv 1678 . . . . . . . 8  |-  F/ z
ph
76sb8e 2141 . . . . . . 7  |-  ( E. y ph  <->  E. z [ z  /  y ] ph )
87sbcbii 3384 . . . . . 6  |-  ( [. A  /  x ]. E. y ph  <->  [. A  /  x ]. E. z [ z  /  y ] ph )
98imbi2i 312 . . . . 5  |-  ( ( y  =  z  ->  [. A  /  x ]. E. y ph )  <->  ( y  =  z  ->  [. A  /  x ]. E. z [ z  /  y ] ph ) )
109bicomi 202 . . . 4  |-  ( ( y  =  z  ->  [. A  /  x ]. E. z [ z  /  y ] ph ) 
<->  ( y  =  z  ->  [. A  /  x ]. E. y ph )
)
1110pm5.74ri 246 . . 3  |-  ( y  =  z  ->  ( [. A  /  x ]. E. z [ z  /  y ] ph  <->  [. A  /  x ]. E. y ph ) )
12 nfs1v 2157 . . . . . 6  |-  F/ y [ z  /  y ] ph
131, 12nfsbc 3346 . . . . 5  |-  F/ y
[. A  /  x ]. [ z  /  y ] ph
14 nfv 1678 . . . . 5  |-  F/ z
[. A  /  x ]. ph
15 sbequ12r 1955 . . . . . 6  |-  ( z  =  y  ->  ( [ z  /  y ] ph  <->  ph ) )
1615sbcbidv 3383 . . . . 5  |-  ( z  =  y  ->  ( [. A  /  x ]. [ z  /  y ] ph  <->  [. A  /  x ]. ph ) )
1713, 14, 16cbvex 1988 . . . 4  |-  ( E. z [. A  /  x ]. [ z  / 
y ] ph  <->  E. y [. A  /  x ]. ph )
1817a1i 11 . . 3  |-  ( y  =  z  ->  ( E. z [. A  /  x ]. [ z  / 
y ] ph  <->  E. y [. A  /  x ]. ph ) )
1911, 18bibi12d 321 . 2  |-  ( y  =  z  ->  (
( [. A  /  x ]. E. z [ z  /  y ] ph  <->  E. z [. A  /  x ]. [ z  / 
y ] ph )  <->  (
[. A  /  x ]. E. y ph  <->  E. y [. A  /  x ]. ph ) ) )
20 sbcex2 3378 . 2  |-  ( [. A  /  x ]. E. z [ z  /  y ] ph  <->  E. z [. A  /  x ]. [ z  /  y ] ph )
215, 19, 20chvar 1975 1  |-  ( [. A  /  x ]. E. y ph  <->  E. y [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374   E.wex 1591   [wsb 1706   F/_wnfc 2608   [.wsbc 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-sbc 3325
This theorem is referenced by:  sbcexfi  30110
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