Theorem reiotacl 4364

Description: Membership law for descriptions. (Contributed by SF, 21-Aug-2011.)

Assertion

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

Distinct variable group:   x,A

Allowed substitution hint:   ph(x)

Proof of Theorem reiotacl

Step	Hyp	Ref	Expression
1		ssrab2 3351	. . 3 |- {x e. A | ph} C_ A
2	1	a1i 10	. 2 |- (E!x e. A ph -> {x e. A | ph} C_ A)
3		reiotacl2 4363	. 2 |- (E!x e. A ph -> (iotax(x e. A /\ ph)) e. {x e. A | ph})
4	2, 3	sseldd 3274	1 |- (E!x e. A ph -> (iotax(x e. A /\ ph)) e. A)

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710  E!wreu 2616  {crab 2618   C_ wss 3257  iotacio 4337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-reu 2621  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339

This theorem is referenced by:  ncfinprop  4474  tfinprop  4489  tccl  6160