Theorem iota5f 31606

Description: A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)

Hypotheses

Ref	Expression
iota5f.1	|- F/ x ph
iota5f.2	|- F/_ x A
iota5f.3	|- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) )

Assertion

Ref	Expression
iota5f	|- ( ( ph /\ A e. V ) -> ( iota x ps ) = A )

Distinct variable group:   x, V

Allowed substitution hints:   ph( x)   ps( x)   A( x)

Proof of Theorem iota5f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iota5f.1	. . . 4 |- F/ x ph
2		iota5f.2	. . . . 5 |- F/_ x A
3	2	nfel1 2779	. . . 4 |- F/ x A e. V
4	1, 3	nfan 1828	. . 3 |- F/ x ( ph /\ A e. V )
5		iota5f.3	. . 3 |- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) )
6	4, 5	alrimi 2082	. 2 |- ( ( ph /\ A e. V ) -> A. x ( ps <-> x = A ) )
7	2	nfeq2 2780	. . . . . 6 |- F/ x y = A
8		eqeq2 2633	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
9	8	bibi2d 332	. . . . . 6 |- ( y = A -> ( ( ps <-> x = y ) <-> ( ps <-> x = A ) ) )
10	7, 9	albid 2090	. . . . 5 |- ( y = A -> ( A. x ( ps <-> x = y ) <-> A. x ( ps <-> x = A ) ) )
11		eqeq2 2633	. . . . 5 |- ( y = A -> ( ( iota x ps ) = y <-> ( iota x ps ) = A ) )
12	10, 11	imbi12d 334	. . . 4 |- ( y = A -> ( ( A. x ( ps <-> x = y ) -> ( iota x ps ) = y ) <-> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) ) )
13		iotaval 5862	. . . 4 |- ( A. x ( ps <-> x = y ) -> ( iota x ps ) = y )
14	12, 13	vtoclg 3266	. . 3 |- ( A e. V -> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) )
15	14	adantl 482	. 2 |- ( ( ph /\ A e. V ) -> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) )
16	6, 15	mpd 15	1 |- ( ( ph /\ A e. V ) -> ( iota x ps ) = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by: (None)