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Metamath Proof Explorer

Theorem sb8iota

Description: Variable substitution in description binder. Compare sb8eu . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 18-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypothesis sb8iota.1 
|- F/ y ph
Assertion sb8iota 
|- ( iota x ph ) = ( iota y [ y / x ] ph )

Proof

Step Hyp Ref Expression
1 sb8iota.1 
 |-  F/ y ph
2 nfv 
 |-  F/ w ( ph <-> x = z )
3 2 sb8 
 |-  ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
4 sbbi 
 |-  ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> [ w / x ] x = z ) )
5 1 nfsb 
 |-  F/ y [ w / x ] ph
6 equsb3 
 |-  ( [ w / x ] x = z <-> w = z )
7 nfv 
 |-  F/ y w = z
8 6 7 nfxfr 
 |-  F/ y [ w / x ] x = z
9 5 8 nfbi 
 |-  F/ y ( [ w / x ] ph <-> [ w / x ] x = z )
10 4 9 nfxfr 
 |-  F/ y [ w / x ] ( ph <-> x = z )
11 nfv 
 |-  F/ w [ y / x ] ( ph <-> x = z )
12 sbequ 
 |-  ( w = y -> ( [ w / x ] ( ph <-> x = z ) <-> [ y / x ] ( ph <-> x = z ) ) )
13 10 11 12 cbvalv1 
 |-  ( A. w [ w / x ] ( ph <-> x = z ) <-> A. y [ y / x ] ( ph <-> x = z ) )
14 equsb3 
 |-  ( [ y / x ] x = z <-> y = z )
15 14 sblbis 
 |-  ( [ y / x ] ( ph <-> x = z ) <-> ( [ y / x ] ph <-> y = z ) )
16 15 albii 
 |-  ( A. y [ y / x ] ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
17 3 13 16 3bitri 
 |-  ( A. x ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
18 17 abbii 
 |-  { z | A. x ( ph <-> x = z ) } = { z | A. y ( [ y / x ] ph <-> y = z ) }
19 18 unieqi 
 |-  U. { z | A. x ( ph <-> x = z ) } = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
20 dfiota2 
 |-  ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
21 dfiota2 
 |-  ( iota y [ y / x ] ph ) = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
22 19 20 21 3eqtr4i 
 |-  ( iota x ph ) = ( iota y [ y / x ] ph )