Theorem iota4an 5101

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 5100	. 2 |- (E!x(ph /\ ps) -> [(iotax(ph /\ ps)) / x](ph /\ ps))
2		iotaex 5099	. . . 4 |- (iotax(ph /\ ps)) e. _V
3		simpl 346	. . . . 5 |- ((ph /\ ps) -> ph)
4	3	sbcth 2458	. . . 4 |- ((iotax(ph /\ ps)) e. _V -> [(iotax(ph /\ ps)) / x]((ph /\ ps) -> ph))
5	2, 4	ax-mp 7	. . 3 |- [(iotax(ph /\ ps)) / x]((ph /\ ps) -> ph)
6		sbcimg 2496	. . . 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	ax-mp 7	. . 3 |- ([(iotax(ph /\ ps)) / x]((ph /\ ps) -> ph) <-> ([(iotax(ph /\ ps)) / x](ph /\ ps) -> [(iotax(ph /\ ps)) / x]ph))
8	5, 7	mpbi 206	. 2 |- ([(iotax(ph /\ ps)) / x](ph /\ ps) -> [(iotax(ph /\ ps)) / x]ph)
9	1, 8	syl 12	1 |- (E!x(ph /\ ps) -> [(iotax(ph /\ ps)) / x]ph)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  [wsbc 1534  E!weu 1771  _Vcvv 2292  iotacio 5087

This theorem is referenced by:  reiota4 5107

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-nul 3445

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-sn 3049  df-pr 3050  df-uni 3178  df-iota 5089

Copyright terms: Public domain