Theorem riotasbc 5813

Description: Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
riotasbc	|- ( E! x e. A ph -> [. ( iota_ x e. A ph ) / x ]. ph )

Proof of Theorem riotasbc

Step	Hyp	Ref	Expression
1		rabssab 3230	. . 3 |- { x e. A | ph } C_ { x | ph }
2		riotacl2 5811	. . 3 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } )
3	1, 2	sselid 3140	. 2 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x | ph } )
4		df-sbc 2952	. 2 |- ( [. ( iota_ x e. A ph ) / x ]. ph <-> ( iota_ x e. A ph ) e. { x | ph } )
5	3, 4	sylibr 133	1 |- ( E! x e. A ph -> [. ( iota_ x e. A ph ) / x ]. ph )

Syntax hints:   -> wi 4   e. wcel 2136   {cab 2151   E!wreu 2446   {crab 2448   [.wsbc 2951   iota_crio 5797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-uni 3790  df-iota 5153  df-riota 5798

This theorem is referenced by:  riotass2  5824  riotass  5825  cjth  10788