Theorem iota4an 5564

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 5563	. 2 |- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) )
2		iotaex 5562	. . . 4 |- ( iota x ( ph /\ ps ) ) e. _V
3		simpl 459	. . . . 5 |- ( ( ph /\ ps ) -> ph )
4	3	sbcth 3281	. . . 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 3308	. . . 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 212	. 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 188   /\ wa 371   e. wcel 1886   E!weu 2298   _Vcvv 3044   [.wsbc 3266   iotacio 5543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-nul 4533

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-sn 3968  df-pr 3970  df-uni 4198  df-iota 5545

This theorem is referenced by: (None)