Theorem cbviotavw 5341

Description: Change bound variables in a description binder. Version of cbviotav 5342 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 2365	. . . . 5 |- { x | ph } = { y | ps }
3	2	eqeq1i 2246	. . . 4 |- ( { x | ph } = { z } <-> { y | ps } = { z } )
4	3	abbii 2354	. . 3 |- { z | { x | ph } = { z } } = { z | { y | ps } = { z } }
5	4	unieqi 3943	. 2 |- U. { z | { x | ph } = { z } } = U. { z | { y | ps } = { z } }
6		df-iota 5335	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
7		df-iota 5335	. 2 |- ( iota y ps ) = U. { z | { y | ps } = { z } }
8	5, 6, 7	3eqtr4i 2269	1 |- ( iota x ph ) = ( iota y ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   {cab 2224   {csn 3708   U.cuni 3933   iotacio 5333

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

This theorem is referenced by:  cbvriotavw  6043