Theorem iota2d 5258

Description: A condition that allows us to represent "the unique element such that ph " with a class expression A. (Contributed by NM, 30-Dec-2014.)

Hypotheses

Ref	Expression
iota2df.1	|- ( ph -> B e. V )
iota2df.2	|- ( ph -> E! x ps )
iota2df.3	|- ( ( ph /\ x = B ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
iota2d	|- ( ph -> ( ch <-> ( iota x ps ) = B ) )

Distinct variable groups:   x, B   ph, x   ch, x

Allowed substitution hints:   ps( x)   V( x)

Proof of Theorem iota2d

Step	Hyp	Ref	Expression
1		iota2df.1	. 2 |- ( ph -> B e. V )
2		iota2df.2	. 2 |- ( ph -> E! x ps )
3		iota2df.3	. 2 |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )
4		nfv 1551	. 2 |- F/ x ph
5		nfvd 1552	. 2 |- ( ph -> F/ x ch )
6		nfcvd 2349	. 2 |- ( ph -> F/_ x B )
7	1, 2, 3, 4, 5, 6	iota2df 5257	1 |- ( ph -> ( ch <-> ( iota x ps ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E!weu 2054   e. wcel 2176   iotacio 5230

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-sn 3639  df-pr 3640  df-uni 3851  df-iota 5232

This theorem is referenced by: (None)