Theorem riota1 5996

Description: Property of restricted iota. Compare iota1 5303. (Contributed by Mario Carneiro, 15-Oct-2016.)

Assertion

Ref	Expression
riota1	|- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem riota1

Step	Hyp	Ref	Expression
1		df-reu 2516	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iota1 5303	. . 3 |- ( E! x ( x e. A /\ ph ) -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
3	1, 2	sylbi 121	. 2 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
4		df-riota 5976	. . 3 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5	4	eqeq1i 2238	. 2 |- ( ( iota_ x e. A ph ) = x <-> ( iota x ( x e. A /\ ph ) ) = x )
6	3, 5	bitr4di 198	1 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   E!weu 2078   e. wcel 2201   E!wreu 2511   iotacio 5286   iota_crio 5975

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rex 2515  df-reu 2516  df-v 2803  df-sbc 3031  df-un 3203  df-sn 3676  df-pr 3677  df-uni 3895  df-iota 5288  df-riota 5976

This theorem is referenced by:  supelti  7206  oddpwdclemdvds  12765  oddpwdclemndvds  12766