Theorem pm11.58 36810

Description: Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm11.58	|- ( E. x ph <-> E. x E. y ( ph /\ [ y / x ] ph ) )

Distinct variable group:   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem pm11.58

Step	Hyp	Ref	Expression
1		19.8a 1955	. . . . 5 |- ( ph -> E. x ph )
2		nfv 1769	. . . . . 6 |- F/ y ph
3	2	sb8e 2274	. . . . 5 |- ( E. x ph <-> E. y [ y / x ] ph )
4	1, 3	sylib 201	. . . 4 |- ( ph -> E. y [ y / x ] ph )
5	4	pm4.71i 644	. . 3 |- ( ph <-> ( ph /\ E. y [ y / x ] ph ) )
6		19.42v 1842	. . 3 |- ( E. y ( ph /\ [ y / x ] ph ) <-> ( ph /\ E. y [ y / x ] ph ) )
7	5, 6	bitr4i 260	. 2 |- ( ph <-> E. y ( ph /\ [ y / x ] ph ) )
8	7	exbii 1726	1 |- ( E. x ph <-> E. x E. y ( ph /\ [ y / x ] ph ) )

Syntax hints:   <-> wb 189   /\ wa 376   E.wex 1671   [wsb 1805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806

This theorem is referenced by: (None)