Theorem pm11.58 31537

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 1862	. . . . 5 |- ( ph -> E. x ph )
2		nfv 1712	. . . . . 6 |- F/ y ph
3	2	sb8e 2170	. . . . 5 |- ( E. x ph <-> E. y [ y / x ] ph )
4	1, 3	sylib 196	. . . 4 |- ( ph -> E. y [ y / x ] ph )
5	4	pm4.71i 630	. . 3 |- ( ph <-> ( ph /\ E. y [ y / x ] ph ) )
6		19.42v 1780	. . 3 |- ( E. y ( ph /\ [ y / x ] ph ) <-> ( ph /\ E. y [ y / x ] ph ) )
7	5, 6	bitr4i 252	. 2 |- ( ph <-> E. y ( ph /\ [ y / x ] ph ) )
8	7	exbii 1672	1 |- ( E. x ph <-> E. x E. y ( ph /\ [ y / x ] ph ) )

Syntax hints:   <-> wb 184   /\ wa 367   E.wex 1617   [wsb 1744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004

This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745

This theorem is referenced by: (None)