Theorem riotasbc 5699

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 3150	. . 3 |- { x e. A | ph } C_ { x | ph }
2		riotacl2 5697	. . 3 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } )
3	1, 2	sseldi 3061	. 2 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x | ph } )
4		df-sbc 2879	. 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 1463   {cab 2101   E!wreu 2392   {crab 2394   [.wsbc 2878   iota_crio 5683

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500  df-uni 3703  df-iota 5046  df-riota 5684

This theorem is referenced by:  riotass2  5710  riotass  5711  cjth  10511