Theorem pm11.58 38590

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 2052	. . . . 5 |- ( ph -> E. x ph )
2		nfv 1843	. . . . . 6 |- F/ y ph
3	2	sb8e 2425	. . . . 5 |- ( E. x ph <-> E. y [ y / x ] ph )
4	1, 3	sylib 208	. . . 4 |- ( ph -> E. y [ y / x ] ph )
5	4	pm4.71i 664	. . 3 |- ( ph <-> ( ph /\ E. y [ y / x ] ph ) )
6		19.42v 1918	. . 3 |- ( E. y ( ph /\ [ y / x ] ph ) <-> ( ph /\ E. y [ y / x ] ph ) )
7	5, 6	bitr4i 267	. 2 |- ( ph <-> E. y ( ph /\ [ y / x ] ph ) )
8	7	exbii 1774	1 |- ( E. x ph <-> E. x E. y ( ph /\ [ y / x ] ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   [wsb 1880

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)