MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcnestgf Unicode version

Theorem sbcnestgf 3128
Description: Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
sbcnestgf  |-  ( ( A  e.  V  /\  A. y F/ x ph )  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) )

Proof of Theorem sbcnestgf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2993 . . . . 5  |-  ( z  =  A  ->  ( [. z  /  x ]. [. B  /  y ]. ph  <->  [. A  /  x ]. [. B  /  y ]. ph ) )
2 csbeq1 3084 . . . . . 6  |-  ( z  =  A  ->  [_ z  /  x ]_ B  = 
[_ A  /  x ]_ B )
3 dfsbcq 2993 . . . . . 6  |-  ( [_ z  /  x ]_ B  =  [_ A  /  x ]_ B  ->  ( [. [_ z  /  x ]_ B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) )
42, 3syl 15 . . . . 5  |-  ( z  =  A  ->  ( [. [_ z  /  x ]_ B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
51, 4bibi12d 312 . . . 4  |-  ( z  =  A  ->  (
( [. z  /  x ]. [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph )  <->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) ) )
65imbi2d 307 . . 3  |-  ( z  =  A  ->  (
( A. y F/ x ph  ->  ( [. z  /  x ]. [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )  <->  ( A. y F/ x ph  ->  (
[. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) ) ) )
7 vex 2791 . . . . 5  |-  z  e. 
_V
87a1i 10 . . . 4  |-  ( A. y F/ x ph  ->  z  e.  _V )
9 csbeq1a 3089 . . . . . 6  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
10 dfsbcq 2993 . . . . . 6  |-  ( B  =  [_ z  /  x ]_ B  ->  ( [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )
119, 10syl 15 . . . . 5  |-  ( x  =  z  ->  ( [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )
1211adantl 452 . . . 4  |-  ( ( A. y F/ x ph  /\  x  =  z )  ->  ( [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  / 
y ]. ph ) )
13 nfnf1 1757 . . . . 5  |-  F/ x F/ x ph
1413nfal 1766 . . . 4  |-  F/ x A. y F/ x ph
15 nfa1 1756 . . . . 5  |-  F/ y A. y F/ x ph
16 nfcsb1v 3113 . . . . . 6  |-  F/_ x [_ z  /  x ]_ B
1716a1i 10 . . . . 5  |-  ( A. y F/ x ph  ->  F/_ x [_ z  /  x ]_ B )
18 sp 1716 . . . . 5  |-  ( A. y F/ x ph  ->  F/ x ph )
1915, 17, 18nfsbcd 3011 . . . 4  |-  ( A. y F/ x ph  ->  F/ x [. [_ z  /  x ]_ B  / 
y ]. ph )
208, 12, 14, 19sbciedf 3026 . . 3  |-  ( A. y F/ x ph  ->  (
[. z  /  x ]. [. B  /  y ]. ph  <->  [. [_ z  /  x ]_ B  /  y ]. ph ) )
216, 20vtoclg 2843 . 2  |-  ( A  e.  V  ->  ( A. y F/ x ph  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) ) )
2221imp 418 1  |-  ( ( A  e.  V  /\  A. y F/ x ph )  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  / 
y ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  csbnestgf  3129  sbcnestg  3130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
  Copyright terms: Public domain W3C validator