Theorem riotasbc 5824

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 3235	. . 3 |- { x e. A | ph } C_ { x | ph }
2		riotacl2 5822	. . 3 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } )
3	1, 2	sselid 3145	. 2 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x | ph } )
4		df-sbc 2956	. 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 2141   {cab 2156   E!wreu 2450   {crab 2452   [.wsbc 2955   iota_crio 5808

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-uni 3797  df-iota 5160  df-riota 5809

This theorem is referenced by:  riotass2  5835  riotass  5836  cjth  10810