Theorem reiota4 5107

Description: Substitution law for descriptions. Compare reuuni4 3813.

Assertion

Ref	Expression
reiota4	|- (E!x e. A ph -> [(iotax(x e. A /\ ph)) / x]ph)

Proof of Theorem reiota4

Step	Hyp	Ref	Expression
1		iota4an 5101	. 2 |- (E!x(ph /\ x e. A) -> [(iotax(ph /\ x e. A)) / x]ph)
2		df-reu 2111	. . 3 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
3		ancom 482	. . . 4 |- ((x e. A /\ ph) <-> (ph /\ x e. A))
4	3	eubii 1780	. . 3 |- (E!x(x e. A /\ ph) <-> E!x(ph /\ x e. A))
5	2, 4	bitri 190	. 2 |- (E!x e. A ph <-> E!x(ph /\ x e. A))
6		iotabi 5094	. . . 4 |- (A.x((x e. A /\ ph) <-> (ph /\ x e. A)) -> (iotax(x e. A /\ ph)) = (iotax(ph /\ x e. A)))
7	6, 3	mpg 1332	. . 3 |- (iotax(x e. A /\ ph)) = (iotax(ph /\ x e. A))
8		dfsbcq 2455	. . 3 |- ((iotax(x e. A /\ ph)) = (iotax(ph /\ x e. A)) -> ([(iotax(x e. A /\ ph)) / x]ph <-> [(iotax(ph /\ x e. A)) / x]ph))
9	7, 8	ax-mp 7	. 2 |- ([(iotax(x e. A /\ ph)) / x]ph <-> [(iotax(ph /\ x e. A)) / x]ph)
10	1, 5, 9	3imtr4i 236	1 |- (E!x e. A ph -> [(iotax(x e. A /\ ph)) / x]ph)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  E!weu 1771  E!wreu 2107  iotacio 5087

This theorem is referenced by:  reiotass2 5111  riota4 5577

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-reu 2111  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

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