Theorem sbcnestg 2583

Description: Nest the composition of two substitutions.

Assertion

Ref	Expression
sbcnestg	|- ((A e. R /\ A.x B e. S) -> ([A / x][B / y]ph <-> [[_A / x]_B / y]ph))

Distinct variable groups:   ph,x   x,y

Proof of Theorem sbcnestg

Step	Hyp	Ref	Expression
1		hba1 1350	. . . . 5 |- (A.x B e. _V -> A.xA.x B e. _V)
2		sbccsb2g 2566	. . . . . 6 |- (B e. _V -> ([B / y]ph <-> B e. [_B / y]_{y | ph}))
3	2	a4s 1330	. . . . 5 |- (A.x B e. _V -> ([B / y]ph <-> B e. [_B / y]_{y | ph}))
4	1, 3	sbcbid 2504	. . . 4 |- ((A.x B e. _V /\ A e. R) -> ([A / x][B / y]ph <-> [A / x]B e. [_B / y]_{y | ph}))
5	4	ancoms 484	. . 3 |- ((A e. R /\ A.x B e. _V) -> ([A / x][B / y]ph <-> [A / x]B e. [_B / y]_{y | ph}))
6		sbcel12g 2552	. . . 4 |- (A e. R -> ([A / x]B e. [_B / y]_{y | ph} <-> [_A / x]_B e. [_A / x]_[_B / y]_{y | ph}))
7	6	adantr 425	. . 3 |- ((A e. R /\ A.x B e. _V) -> ([A / x]B e. [_B / y]_{y | ph} <-> [_A / x]_B e. [_A / x]_[_B / y]_{y | ph}))
8		csbnestg 2581	. . . . 5 |- ((A e. R /\ A.x B e. _V) -> [_A / x]_[_B / y]_{y | ph} = [_[_A / x]_B / y]_{y | ph})
9	8	eleq2d 1964	. . . 4 |- ((A e. R /\ A.x B e. _V) -> ([_A / x]_B e. [_A / x]_[_B / y]_{y | ph} <-> [_A / x]_B e. [_[_A / x]_B / y]_{y | ph}))
10		csbexg 2548	. . . . 5 |- ((A e. R /\ A.x B e. _V) -> [_A / x]_B e. _V)
11		sbccsb2g 2566	. . . . 5 |- ([_A / x]_B e. _V -> ([[_A / x]_B / y]ph <-> [_A / x]_B e. [_[_A / x]_B / y]_{y | ph}))
12	10, 11	syl 12	. . . 4 |- ((A e. R /\ A.x B e. _V) -> ([[_A / x]_B / y]ph <-> [_A / x]_B e. [_[_A / x]_B / y]_{y | ph}))
13	9, 12	bitr4d 590	. . 3 |- ((A e. R /\ A.x B e. _V) -> ([_A / x]_B e. [_A / x]_[_B / y]_{y | ph} <-> [[_A / x]_B / y]ph))
14	5, 7, 13	3bitrd 603	. 2 |- ((A e. R /\ A.x B e. _V) -> ([A / x][B / y]ph <-> [[_A / x]_B / y]ph))
15		elisset 2299	. . 3 |- (B e. S -> B e. _V)
16	15	alimi 1338	. 2 |- (A.x B e. S -> A.x B e. _V)
17	14, 16	sylan2 500	1 |- ((A e. R /\ A.x B e. S) -> ([A / x][B / y]ph <-> [[_A / x]_B / y]ph))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300  [wsbc 1534  {cab 1871  _Vcvv 2292  [_csb 2540

This theorem is referenced by:  sbcco3g 2586

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541

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