Theorem iota4an 4359

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 4358	. 2 |- (E!x(ph /\ ps) -> [.(iotax(ph /\ ps)) / x ].(ph /\ ps))
2		iotaex 4357	. . . 4 |- (iotax(ph /\ ps)) e. _V
3		simpl 443	. . . . 5 |- ((ph /\ ps) -> ph)
4	3	sbcth 3061	. . . 4 |- ((iotax(ph /\ ps)) e. _V -> [.(iotax(ph /\ ps)) / x ].((ph /\ ps) -> ph))
5	2, 4	ax-mp 5	. . 3 |- [.(iotax(ph /\ ps)) / x ].((ph /\ ps) -> ph)
6		sbcimg 3088	. . . 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 5	. . 3 |- ( [.(iotax(ph /\ ps)) / x ].((ph /\ ps) -> ph) <-> ( [.(iotax(ph /\ ps)) / x ].(ph /\ ps) -> [.(iotax(ph /\ ps)) / x ].ph))
8	5, 7	mpbi 199	. 2 |- ( [.(iotax(ph /\ ps)) / x ].(ph /\ ps) -> [.(iotax(ph /\ ps)) / x ].ph)
9	1, 8	syl 15	1 |- (E!x(ph /\ ps) -> [.(iotax(ph /\ ps)) / x ].ph)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   e. wcel 1710  E!weu 2204  _Vcvv 2860   [.wsbc 3047  iotacio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by: (None)