Theorem cbvriotavw 5982

Description: Change bound variable in a restricted description binder. Version of cbvriotav 5984 with a disjoint variable condition. (Contributed by NM, 18-Mar-2013.) (Revised by GG, 30-Sep-2024.)

Hypothesis

Ref	Expression
cbvriotavw.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvriotavw	|- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

Distinct variable groups:   x, y, A   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem cbvriotavw

Step	Hyp	Ref	Expression
1		eleq1w 2292	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
2		cbvriotavw.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	anbi12d 473	. . 3 |- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) )
4	3	cbviotavw 5292	. 2 |- ( iota x ( x e. A /\ ph ) ) = ( iota y ( y e. A /\ ps ) )
5		df-riota 5971	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
6		df-riota 5971	. 2 |- ( iota_ y e. A ps ) = ( iota y ( y e. A /\ ps ) )
7	4, 5, 6	3eqtr4i 2262	1 |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   iotacio 5284   iota_crio 5970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-uni 3894  df-iota 5286  df-riota 5971

This theorem is referenced by:  uspgredg2v  16075  usgredg2v  16078