Theorem sbcnestg 2009

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 979	. . . . 5 |- (A.x B e. V -> A.xA.x B e. V)
2		sbccsb2g 1994	. . . . . 6 |- (B e. V -> ([B / y]ph <-> B e. [_B / y]_{y | ph}))
3	2	a4s 960	. . . . 5 |- (A.x B e. V -> ([B / y]ph <-> B e. [_B / y]_{y | ph}))
4	1, 3	sbcbid 1947	. . . 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 436	. . 3 |- ((A e. R /\ A.x B e. V) -> ([A / x][B / y]ph <-> [A / x]B e. [_B / y]_{y | ph}))
6		sbcel12g 1982	. . . 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 389	. . 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 2007	. . . . 5 |- ((A e. R /\ A.x B e. V) -> [_A / x]_[_B / y]_{y | ph} = [_[_A / x]_B / y]_{y | ph})
9	8	eleq2d 1517	. . . 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 1979	. . . . 5 |- ((A e. R /\ A.x B e. V) -> [_A / x]_B e. V)
11		sbccsb2g 1994	. . . . 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 10	. . . 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 529	. . 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 542	. 2 |- ((A e. R /\ A.x B e. V) -> ([A / x][B / y]ph <-> [[_A / x]_B / y]ph))
15		elisset 1792	. . 3 |- (B e. S -> B e. V)
16	15	19.20i 968	. 2 |- (A.x B e. S -> A.x B e. V)
17	14, 16	sylan2 451	1 |- ((A e. R /\ A.x B e. S) -> ([A / x][B / y]ph <-> [[_A / x]_B / y]ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950   e. wcel 1105  [wsbc 1153  {cab 1440  Vcvv 1786  [_csb 1972

This theorem is referenced by:  sbcco3g 2012

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913  df-csb 1973

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