Theorem sbaniota 38636

Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
sbaniota	|- ( E! x ph -> ( E. x ( ph /\ ps ) <-> [. ( iota x ph ) / x ]. ps ) )

Proof of Theorem sbaniota

Step	Hyp	Ref	Expression
1		eupickbi 2539	. 2 |- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) )
2		sbiota1 38635	. 2 |- ( E! x ph -> ( A. x ( ph -> ps ) <-> [. ( iota x ph ) / x ]. ps ) )
3	1, 2	bitrd 268	1 |- ( E! x ph -> ( E. x ( ph /\ ps ) <-> [. ( iota x ph ) / x ]. ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   E!weu 2470   [.wsbc 3435   iotacio 5849

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-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by: (None)