Theorem iota2d 5359

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 1581	. 2 |- F/ x ph
5		nfvd 1582	. 2 |- ( ph -> F/ x ch )
6		nfcvd 2393	. 2 |- ( ph -> F/_ x B )
7	1, 2, 3, 4, 5, 6	iota2df 5358	1 |- ( ph -> ( ch <-> ( iota x ps ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E!weu 2086   e. wcel 2209   iotacio 5330

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3711  df-pr 3712  df-uni 3931  df-iota 5332

This theorem is referenced by: (None)