Theorem pm14.12 38622

Description: Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
pm14.12	|- ( E! x ph -> A. x A. y ( ( ph /\ [. y / x ]. ph ) -> x = y ) )

Distinct variable groups:   ph, y   x, y

Allowed substitution hint:   ph( x)

Proof of Theorem pm14.12

Step	Hyp	Ref	Expression
1		eumo 2499	. 2 |- ( E! x ph -> E* x ph )
2		nfv 1843	. . . 4 |- F/ y ph
3	2	mo3 2507	. . 3 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
4		sbsbc 3439	. . . . . 6 |- ( [ y / x ] ph <-> [. y / x ]. ph )
5	4	anbi2i 730	. . . . 5 |- ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ [. y / x ]. ph ) )
6	5	imbi1i 339	. . . 4 |- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ [. y / x ]. ph ) -> x = y ) )
7	6	2albii 1748	. . 3 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> A. x A. y ( ( ph /\ [. y / x ]. ph ) -> x = y ) )
8	3, 7	bitri 264	. 2 |- ( E* x ph <-> A. x A. y ( ( ph /\ [. y / x ]. ph ) -> x = y ) )
9	1, 8	sylib 208	1 |- ( E! x ph -> A. x A. y ( ( ph /\ [. y / x ]. ph ) -> x = y ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   [wsb 1880   E!weu 2470   E*wmo 2471   [.wsbc 3435

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by:  pm14.24  38633