Theorem sbcnestg 2034

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 1001	. . . . 5 |- (A.x B e. V -> A.xA.x B e. V)
2		sbccsb2g 2019	. . . . . 6 |- (B e. V -> ([B / y]ph <-> B e. [_B / y]_{y | ph}))
3	2	a4s 982	. . . . 5 |- (A.x B e. V -> ([B / y]ph <-> B e. [_B / y]_{y | ph}))
4	1, 3	sbcbid 1972	. . . 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 2007	. . . 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 2032	. . . . 5 |- ((A e. R /\ A.x B e. V) -> [_A / x]_[_B / y]_{y | ph} = [_[_A / x]_B / y]_{y | ph})
9	8	eleq2d 1538	. . . 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 2004	. . . . 5 |- ((A e. R /\ A.x B e. V) -> [_A / x]_B e. V)
11		sbccsb2g 2019	. . . . 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 530	. . 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 543	. 2 |- ((A e. R /\ A.x B e. V) -> ([A / x][B / y]ph <-> [[_A / x]_B / y]ph))
15		elisset 1813	. . 3 |- (B e. S -> B e. V)
16	15	19.20i 990	. 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 952   e. wcel 956  [wsbc 1168  {cab 1461  Vcvv 1807  [_csb 1997

This theorem is referenced by:  sbcco3g 2037

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938  df-csb 1998

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