Theorem iota4an 5870

Description: Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
iota4an	|- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )

Proof of Theorem iota4an

Step	Hyp	Ref	Expression
1		iota4 5869	. 2 |- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) )
2		iotaex 5868	. . . 4 |- ( iota x ( ph /\ ps ) ) e. _V
3		simpl 473	. . . . 5 |- ( ( ph /\ ps ) -> ph )
4	3	sbcth 3450	. . . 4 |- ( ( iota x ( ph /\ ps ) ) e. _V -> [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) )
5	2, 4	ax-mp 5	. . 3 |- [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph )
6		sbcimg 3477	. . . 4 |- ( ( iota x ( ph /\ ps ) ) e. _V -> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) <-> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) ) )
7	2, 6	ax-mp 5	. . 3 |- ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) <-> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) )
8	5, 7	mpbi 220	. 2 |- ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )
9	1, 8	syl 17	1 |- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   E!weu 2470   _Vcvv 3200   [.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  ax-nul 4789

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by: (None)