Theorem iota4an 4829

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

Assertion

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

Proof of Theorem iota4an

Step	Hyp	Ref	Expression
1		iota4 4828	. 2 |- (E!x(ph /\ ps) -> [.( iotax(ph /\ ps)) / x ].(ph /\ ps))
2		euiotaex 4826	. . . 4 |- (E!x(ph /\ ps) -> ( iotax(ph /\ ps)) e. _V)
3		simpl 102	. . . . 5 |- ((ph /\ ps) -> ph)
4	3	sbcth 2771	. . . 4 |- (( iotax(ph /\ ps)) e. _V -> [.( iotax(ph /\ ps)) / x ].((ph /\ ps) -> ph))
5	2, 4	syl 14	. . 3 |- (E!x(ph /\ ps) -> [.( iotax(ph /\ ps)) / x ].((ph /\ ps) -> ph))
6		sbcimg 2798	. . . 4 |- (( iotax(ph /\ ps)) e. _V -> ( [.( iotax(ph /\ ps)) / x ].((ph /\ ps) -> ph) <-> ( [.( iotax(ph /\ ps)) / x ].(ph /\ ps) -> [.( iotax(ph /\ ps)) / x ].ph)))
7	2, 6	syl 14	. . 3 |- (E!x(ph /\ ps) -> ( [.( iotax(ph /\ ps)) / x ].((ph /\ ps) -> ph) <-> ( [.( iotax(ph /\ ps)) / x ].(ph /\ ps) -> [.( iotax(ph /\ ps)) / x ].ph)))
8	5, 7	mpbid 135	. 2 |- (E!x(ph /\ ps) -> ( [.( iotax(ph /\ ps)) / x ].(ph /\ ps) -> [.( iotax(ph /\ ps)) / x ].ph))
9	1, 8	mpd 13	1 |- (E!x(ph /\ ps) -> [.( iotax(ph /\ ps)) / x ].ph)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1390  E!weu 1897   _Vcvv 2551   [.wsbc 2758   iotacio 4808

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810

This theorem is referenced by: (None)