Theorem cbvriotavw 6043

Description: Change bound variable in a restricted description binder. Version of cbvriotav 6045 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 2299	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
2		cbvriotavw.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	anbi12d 477	. . 3 |- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) )
4	3	cbviotavw 5341	. 2 |- ( iota x ( x e. A /\ ph ) ) = ( iota y ( y e. A /\ ps ) )
5		df-riota 6032	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
6		df-riota 6032	. 2 |- ( iota_ y e. A ps ) = ( iota y ( y e. A /\ ps ) )
7	4, 5, 6	3eqtr4i 2269	1 |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   iotacio 5333   iota_crio 6031

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-uni 3934  df-iota 5335  df-riota 6032

This theorem is referenced by:  uspgredg2v  16445  usgredg2v  16448