Theorem sbcnestg 2038

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 1003	. . . . 5 |- (A.x B e. V -> A.xA.x B e. V)
2		sbccsb2g 2023	. . . . . 6 |- (B e. V -> ([B / y]ph <-> B e. [_B / y]_{y | ph}))
3	2	a4s 984	. . . . 5 |- (A.x B e. V -> ([B / y]ph <-> B e. [_B / y]_{y | ph}))
4	1, 3	sbcbid 1976	. . . 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 2011	. . . 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 2036	. . . . 5 |- ((A e. R /\ A.x B e. V) -> [_A / x]_[_B / y]_{y | ph} = [_[_A / x]_B / y]_{y | ph})
9	8	eleq2d 1541	. . . 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 2008	. . . . 5 |- ((A e. R /\ A.x B e. V) -> [_A / x]_B e. V)
11		sbccsb2g 2023	. . . . 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 531	. . 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 544	. 2 |- ((A e. R /\ A.x B e. V) -> ([A / x][B / y]ph <-> [[_A / x]_B / y]ph))
15		elisset 1817	. . 3 |- (B e. S -> B e. V)
16	15	19.20i 992	. 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 954   e. wcel 958  [wsbc 1170  {cab 1463  Vcvv 1811  [_csb 2001

This theorem is referenced by:  sbcco3g 2041

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

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