Theorem sbcnestg 2089

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 1044	. . . . 5 |- (A.x B e. V -> A.xA.x B e. V)
2		sbccsb2g 2074	. . . . . 6 |- (B e. V -> ([B / y]ph <-> B e. [_B / y]_{y | ph}))
3	2	a4s 1025	. . . . 5 |- (A.x B e. V -> ([B / y]ph <-> B e. [_B / y]_{y | ph}))
4	1, 3	sbcbid 2026	. . . 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 447	. . 3 |- ((A e. R /\ A.x B e. V) -> ([A / x][B / y]ph <-> [A / x]B e. [_B / y]_{y | ph}))
6		sbcel12g 2062	. . . 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 398	. . 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 2087	. . . . 5 |- ((A e. R /\ A.x B e. V) -> [_A / x]_[_B / y]_{y | ph} = [_[_A / x]_B / y]_{y | ph})
9	8	eleq2d 1588	. . . 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 2059	. . . . 5 |- ((A e. R /\ A.x B e. V) -> [_A / x]_B e. V)
11		sbccsb2g 2074	. . . . 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 542	. . 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 555	. 2 |- ((A e. R /\ A.x B e. V) -> ([A / x][B / y]ph <-> [[_A / x]_B / y]ph))
15		elisset 1864	. . 3 |- (B e. S -> B e. V)
16	15	19.20i 1033	. 2 |- (A.x B e. S -> A.x B e. V)
17	14, 16	sylan2 462	1 |- ((A e. R /\ A.x B e. S) -> ([A / x][B / y]ph <-> [[_A / x]_B / y]ph))

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230  A.wal 995   e. wcel 999  [wsbc 1212  {cab 1509  Vcvv 1858  [_csb 2051

This theorem is referenced by:  sbcco3g 2092

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-v 1859  df-sbc 1989  df-csb 2052

Copyright terms: Public domain