Theorem iota4

Description: Theorem *14.22 in WhiteheadRussell p. 190. (Contributed by Andrew Salmon, 12-Jul-2011)

	Ref	Expression
Assertion	iota4	|- ( E! x ph -> [. ( iota x ph ) / x ]. ph )

Proof

Step	Hyp	Ref	Expression
1		eu6	|- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		biimpr	|- ( ( ph <-> x = z ) -> ( x = z -> ph ) )
3	2	alimi	|- ( A. x ( ph <-> x = z ) -> A. x ( x = z -> ph ) )
4		sb6	|- ( [ z / x ] ph <-> A. x ( x = z -> ph ) )
5	3 4	sylibr	|- ( A. x ( ph <-> x = z ) -> [ z / x ] ph )
6		iotaval	|- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
7	6	eqcomd	|- ( A. x ( ph <-> x = z ) -> z = ( iota x ph ) )
8		dfsbcq2	|- ( z = ( iota x ph ) -> ( [ z / x ] ph <-> [. ( iota x ph ) / x ]. ph ) )
9	7 8	syl	|- ( A. x ( ph <-> x = z ) -> ( [ z / x ] ph <-> [. ( iota x ph ) / x ]. ph ) )
10	5 9	mpbid	|- ( A. x ( ph <-> x = z ) -> [. ( iota x ph ) / x ]. ph )
11	10	exlimiv	|- ( E. z A. x ( ph <-> x = z ) -> [. ( iota x ph ) / x ]. ph )
12	1 11	sylbi	|- ( E! x ph -> [. ( iota x ph ) / x ]. ph )