Theorem riotasbc 5503

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 3081	. . 3 |- { x e. A | ph } C_ { x | ph }
2		riotacl2 5501	. . 3 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } )
3	1, 2	sseldi 2997	. 2 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x | ph } )
4		df-sbc 2816	. 2 |- ( [. ( iota_ x e. A ph ) / x ]. ph <-> ( iota_ x e. A ph ) e. { x | ph } )
5	3, 4	sylibr 132	1 |- ( E! x e. A ph -> [. ( iota_ x e. A ph ) / x ]. ph )

Syntax hints:   -> wi 4   e. wcel 1433   {cab 2067   E!wreu 2350   {crab 2352   [.wsbc 2815   iota_crio 5487

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887  df-riota 5488

This theorem is referenced by:  riotass2  5514  riotass  5515  cjth  9733