Theorem riota1 5680

Description: Property of restricted iota. Compare iota1 5038. (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 2382	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iota1 5038	. . 3 |- ( E! x ( x e. A /\ ph ) -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
3	1, 2	sylbi 120	. 2 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
4		df-riota 5662	. . 3 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5	4	eqeq1i 2107	. 2 |- ( ( iota_ x e. A ph ) = x <-> ( iota x ( x e. A /\ ph ) ) = x )
6	3, 5	syl6bbr 197	1 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   E!weu 1960   E!wreu 2377   iotacio 5022   iota_crio 5661

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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-reu 2382  df-v 2643  df-sbc 2863  df-un 3025  df-sn 3480  df-pr 3481  df-uni 3684  df-iota 5024  df-riota 5662

This theorem is referenced by:  supelti  6804  oddpwdclemdvds  11640  oddpwdclemndvds  11641