Theorem csbabg 2043

Description: Move substitution into a class abstraction.

Assertion

Ref	Expression
csbabg	|- (A e. B -> [_A / x]_{y | ph} = {y | [A / x]ph})

Distinct variable groups:   y,A   x,y

Proof of Theorem csbabg

Step	Hyp	Ref	Expression
1		elisset 1817	. 2 |- (A e. B -> A e. V)
2		visset 1813	. . . . . . 7 |- z e. V
3		sbccomg 1989	. . . . . . 7 |- ((z e. V /\ A e. V) -> ([z / y][A / x]ph <-> [A / x][z / y]ph))
4	2, 3	mpan 695	. . . . . 6 |- (A e. V -> ([z / y][A / x]ph <-> [A / x][z / y]ph))
5		df-clab 1464	. . . . . . 7 |- (z e. {y | ph} <-> [z / y]ph)
6	5	sbcbii 1978	. . . . . 6 |- (A e. V -> ([A / x]z e. {y | ph} <-> [A / x][z / y]ph))
7	4, 6	bitr4d 531	. . . . 5 |- (A e. V -> ([z / y][A / x]ph <-> [A / x]z e. {y | ph}))
8	7	abbidv 1577	. . . 4 |- (A e. V -> {z | [z / y][A / x]ph} = {z | [A / x]z e. {y | ph}})
9		ax-17 971	. . . . 5 |- ([A / x]ph -> A.z[A / x]ph)
10		hbs1 1332	. . . . 5 |- ([z / y][A / x]ph -> A.y[z / y][A / x]ph)
11		sbequ12 1181	. . . . 5 |- (y = z -> ([A / x]ph <-> [z / y][A / x]ph))
12	9, 10, 11	cbvab 1908	. . . 4 |- {y | [A / x]ph} = {z | [z / y][A / x]ph}
13	8, 12	syl5eq 1519	. . 3 |- (A e. V -> {y | [A / x]ph} = {z | [A / x]z e. {y | ph}})
14		df-csb 2002	. . 3 |- [_A / x]_{y | ph} = {z | [A / x]z e. {y | ph}}
15	13, 14	syl6reqr 1526	. 2 |- (A e. V -> [_A / x]_{y | ph} = {y | [A / x]ph})
16	1, 15	syl 10	1 |- (A e. B -> [_A / x]_{y | ph} = {y | [A / x]ph})

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  [wsbc 1170  {cab 1463  Vcvv 1811  [_csb 2001

This theorem is referenced by:  csbopabg 2678

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-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

Copyright terms: Public domain