Theorem cbviotavw 5299

Description: Change bound variables in a description binder. Version of cbviotav 5300 with a disjoint variable condition. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by GG, 30-Sep-2024.)

Hypothesis

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

Assertion

Ref	Expression
cbviotavw	|- ( iota x ph ) = ( iota y ps )

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

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

Proof of Theorem cbviotavw

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviotavw.1	. . . . . 6 |- ( x = y -> ( ph <-> ps ) )
2	1	cbvabv 2357	. . . . 5 |- { x | ph } = { y | ps }
3	2	eqeq1i 2239	. . . 4 |- ( { x | ph } = { z } <-> { y | ps } = { z } )
4	3	abbii 2347	. . 3 |- { z | { x | ph } = { z } } = { z | { y | ps } = { z } }
5	4	unieqi 3908	. 2 |- U. { z | { x | ph } = { z } } = U. { z | { y | ps } = { z } }
6		df-iota 5293	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
7		df-iota 5293	. 2 |- ( iota y ps ) = U. { z | { y | ps } = { z } }
8	5, 6, 7	3eqtr4i 2262	1 |- ( iota x ph ) = ( iota y ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   {cab 2217   {csn 3673   U.cuni 3898   iotacio 5291

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-uni 3899  df-iota 5293

This theorem is referenced by:  cbvriotavw  5992