Theorem cbviotavw 5260

Description: Change bound variables in a description binder. Version of cbviotav 5261 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 2334	. . . . 5 |- { x | ph } = { y | ps }
3	2	eqeq1i 2217	. . . 4 |- ( { x | ph } = { z } <-> { y | ps } = { z } )
4	3	abbii 2325	. . 3 |- { z | { x | ph } = { z } } = { z | { y | ps } = { z } }
5	4	unieqi 3877	. 2 |- U. { z | { x | ph } = { z } } = U. { z | { y | ps } = { z } }
6		df-iota 5254	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
7		df-iota 5254	. 2 |- ( iota y ps ) = U. { z | { y | ps } = { z } }
8	5, 6, 7	3eqtr4i 2240	1 |- ( iota x ph ) = ( iota y ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1375   {cab 2195   {csn 3646   U.cuni 3867   iotacio 5252

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rex 2494  df-uni 3868  df-iota 5254

This theorem is referenced by:  cbvriotavw  5938