Theorem pm11.58 27457

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 1758	. . . . 5 |- ( ph -> E. x ph )
2		nfv 1626	. . . . . 6 |- F/ y ph
3	2	sb8e 2142	. . . . 5 |- ( E. x ph <-> E. y [ y / x ] ph )
4	1, 3	sylib 189	. . . 4 |- ( ph -> E. y [ y / x ] ph )
5	4	pm4.71i 614	. . 3 |- ( ph <-> ( ph /\ E. y [ y / x ] ph ) )
6		19.42v 1924	. . 3 |- ( E. y ( ph /\ [ y / x ] ph ) <-> ( ph /\ E. y [ y / x ] ph ) )
7	5, 6	bitr4i 244	. 2 |- ( ph <-> E. y ( ph /\ [ y / x ] ph ) )
8	7	exbii 1589	1 |- ( E. x ph <-> E. x E. y ( ph /\ [ y / x ] ph ) )

Syntax hints:   <-> wb 177   /\ wa 359   E.wex 1547   [wsb 1655

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656