Theorem cbviotavw 5294

Description: Change bound variables in a description binder. Version of cbviotav 5295 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 2355	. . . . 5 |- { x | ph } = { y | ps }
3	2	eqeq1i 2238	. . . 4 |- ( { x | ph } = { z } <-> { y | ps } = { z } )
4	3	abbii 2346	. . 3 |- { z | { x | ph } = { z } } = { z | { y | ps } = { z } }
5	4	unieqi 3904	. 2 |- U. { z | { x | ph } = { z } } = U. { z | { y | ps } = { z } }
6		df-iota 5288	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
7		df-iota 5288	. 2 |- ( iota y ps ) = U. { z | { y | ps } = { z } }
8	5, 6, 7	3eqtr4i 2261	1 |- ( iota x ph ) = ( iota y ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   {cab 2216   {csn 3670   U.cuni 3894   iotacio 5286

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 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rex 2515  df-uni 3895  df-iota 5288

This theorem is referenced by:  cbvriotavw  5987