Theorem pm14.24 38633

Description: Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
pm14.24	|- ( E! x ph -> A. y ( [. y / x ]. ph <-> y = ( iota x ph ) ) )

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem pm14.24

Step	Hyp	Ref	Expression
1		nfeu1 2480	. . . . 5 |- F/ x E! x ph
2		nfsbc1v 3455	. . . . 5 |- F/ x [. y / x ]. ph
3		pm14.12 38622	. . . . . . . . . 10 |- ( E! x ph -> A. x A. y ( ( ph /\ [. y / x ]. ph ) -> x = y ) )
4	3	19.21bbi 2060	. . . . . . . . 9 |- ( E! x ph -> ( ( ph /\ [. y / x ]. ph ) -> x = y ) )
5	4	ancomsd 470	. . . . . . . 8 |- ( E! x ph -> ( ( [. y / x ]. ph /\ ph ) -> x = y ) )
6	5	expdimp 453	. . . . . . 7 |- ( ( E! x ph /\ [. y / x ]. ph ) -> ( ph -> x = y ) )
7		pm13.13b 38609	. . . . . . . . 9 |- ( ( [. y / x ]. ph /\ x = y ) -> ph )
8	7	ex 450	. . . . . . . 8 |- ( [. y / x ]. ph -> ( x = y -> ph ) )
9	8	adantl 482	. . . . . . 7 |- ( ( E! x ph /\ [. y / x ]. ph ) -> ( x = y -> ph ) )
10	6, 9	impbid 202	. . . . . 6 |- ( ( E! x ph /\ [. y / x ]. ph ) -> ( ph <-> x = y ) )
11	10	ex 450	. . . . 5 |- ( E! x ph -> ( [. y / x ]. ph -> ( ph <-> x = y ) ) )
12	1, 2, 11	alrimd 2084	. . . 4 |- ( E! x ph -> ( [. y / x ]. ph -> A. x ( ph <-> x = y ) ) )
13		iotaval 5862	. . . . 5 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
14	13	eqcomd 2628	. . . 4 |- ( A. x ( ph <-> x = y ) -> y = ( iota x ph ) )
15	12, 14	syl6 35	. . 3 |- ( E! x ph -> ( [. y / x ]. ph -> y = ( iota x ph ) ) )
16		iota4 5869	. . . 4 |- ( E! x ph -> [. ( iota x ph ) / x ]. ph )
17		dfsbcq 3437	. . . 4 |- ( y = ( iota x ph ) -> ( [. y / x ]. ph <-> [. ( iota x ph ) / x ]. ph ) )
18	16, 17	syl5ibrcom 237	. . 3 |- ( E! x ph -> ( y = ( iota x ph ) -> [. y / x ]. ph ) )
19	15, 18	impbid 202	. 2 |- ( E! x ph -> ( [. y / x ]. ph <-> y = ( iota x ph ) ) )
20	19	alrimiv 1855	1 |- ( E! x ph -> A. y ( [. y / x ]. ph <-> y = ( iota x ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E!weu 2470   [.wsbc 3435   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-eu 2474  df-mo 2475  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)