Theorem reiotacl 5106

Description: Membership law for descriptions. Compare reucl 3213.

Assertion

Ref	Expression
reiotacl	|- (E!x e. A ph -> (iotax(x e. A /\ ph)) e. A)

Distinct variable group:   x,A

Proof of Theorem reiotacl

Step	Hyp	Ref	Expression
1		ssrab2 2692	. . 3 |- {x e. A | ph} C_ A
2	1	a1i 8	. 2 |- (E!x e. A ph -> {x e. A | ph} C_ A)
3		reiotacl2 5105	. 2 |- (E!x e. A ph -> (iotax(x e. A /\ ph)) e. {x e. A | ph})
4	2, 3	sseldd 2620	1 |- (E!x e. A ph -> (iotax(x e. A /\ ph)) e. A)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  E!wreu 2107  {crab 2108   C_ wss 2593  iotacio 5087

This theorem is referenced by:  reiotass2 5111  riotacl 5571  grpidclNEW 17118  grpinvclNEW 17127  ringidcl 17150

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-nul 3445

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-sn 3049  df-pr 3050  df-uni 3178  df-iota 5089

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