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Metamath Proof Explorer

Theorem cbvriotaw

Description: Change bound variable in a restricted description binder. Version of cbvriota with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 18-Mar-2013) Avoid ax-13 . (Revised by GG, 26-Jan-2024)

Ref Expression
Hypotheses cbvriotaw.1 
|- F/ y ph
cbvriotaw.2 
|- F/ x ps
cbvriotaw.3 
|- ( x = y -> ( ph <-> ps ) )
Assertion cbvriotaw 
|- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

Proof

Step Hyp Ref Expression
1 cbvriotaw.1 
 |-  F/ y ph
2 cbvriotaw.2 
 |-  F/ x ps
3 cbvriotaw.3 
 |-  ( x = y -> ( ph <-> ps ) )
4 eleq1w 
 |-  ( x = z -> ( x e. A <-> z e. A ) )
5 sbequ12 
 |-  ( x = z -> ( ph <-> [ z / x ] ph ) )
6 4 5 anbi12d 
 |-  ( x = z -> ( ( x e. A /\ ph ) <-> ( z e. A /\ [ z / x ] ph ) ) )
7 nfv 
 |-  F/ z ( x e. A /\ ph )
8 nfv 
 |-  F/ x z e. A
9 nfs1v 
 |-  F/ x [ z / x ] ph
10 8 9 nfan 
 |-  F/ x ( z e. A /\ [ z / x ] ph )
11 6 7 10 cbviotaw 
 |-  ( iota x ( x e. A /\ ph ) ) = ( iota z ( z e. A /\ [ z / x ] ph ) )
12 eleq1w 
 |-  ( z = y -> ( z e. A <-> y e. A ) )
13 2 3 sbhypf 
 |-  ( z = y -> ( [ z / x ] ph <-> ps ) )
14 12 13 anbi12d 
 |-  ( z = y -> ( ( z e. A /\ [ z / x ] ph ) <-> ( y e. A /\ ps ) ) )
15 nfv 
 |-  F/ y z e. A
16 1 nfsbv 
 |-  F/ y [ z / x ] ph
17 15 16 nfan 
 |-  F/ y ( z e. A /\ [ z / x ] ph )
18 nfv 
 |-  F/ z ( y e. A /\ ps )
19 14 17 18 cbviotaw 
 |-  ( iota z ( z e. A /\ [ z / x ] ph ) ) = ( iota y ( y e. A /\ ps ) )
20 11 19 eqtri 
 |-  ( iota x ( x e. A /\ ph ) ) = ( iota y ( y e. A /\ ps ) )
21 df-riota 
 |-  ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
22 df-riota 
 |-  ( iota_ y e. A ps ) = ( iota y ( y e. A /\ ps ) )
23 20 21 22 3eqtr4i 
 |-  ( iota_ x e. A ph ) = ( iota_ y e. A ps )