Theorem riotasbc 5971

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 3312	. . 3 |- { x e. A | ph } C_ { x | ph }
2		riotacl2 5969	. . 3 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } )
3	1, 2	sselid 3222	. 2 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x | ph } )
4		df-sbc 3029	. 2 |- ( [. ( iota_ x e. A ph ) / x ]. ph <-> ( iota_ x e. A ph ) e. { x | ph } )
5	3, 4	sylibr 134	1 |- ( E! x e. A ph -> [. ( iota_ x e. A ph ) / x ]. ph )

Syntax hints:   -> wi 4   e. wcel 2200   {cab 2215   E!wreu 2510   {crab 2512   [.wsbc 3028   iota_crio 5953

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-uni 3889  df-iota 5278  df-riota 5954

This theorem is referenced by:  riotass2  5983  riotass  5984  cjth  11357