Theorem csbieb 2030

Description: Bidirectional conversion between an implicit class substitution hypothesis x = A -> B = C and its explicit substitution equivalent.

Hypotheses

Ref	Expression
csbieb.1	|- A e. V
csbieb.2	|- (y e. C -> A.x y e. C)

Assertion

Ref	Expression
csbieb	|- (A.x(x = A -> B = C) <-> [_A / x]_B = C)

Distinct variable groups:   x,A   y,C   x,y

Proof of Theorem csbieb

Step	Hyp	Ref	Expression
1		csbieb.1	. . . 4 |- A e. V
2		ax-17 971	. . . 4 |- (z e. A -> A.x z e. A)
3	1, 2	hbcsb1 2025	. . 3 |- (z e. [_A / x]_B -> A.x z e. [_A / x]_B)
4		csbieb.2	. . 3 |- (y e. C -> A.x y e. C)
5	3, 4	hbeq 1565	. 2 |- ([_A / x]_B = C -> A.x[_A / x]_B = C)
6		csbeq1a 2006	. . 3 |- (x = A -> B = [_A / x]_B)
7	6	eqeq1d 1483	. 2 |- (x = A -> (B = C <-> [_A / x]_B = C))
8	5, 1, 7	ceqsal 1826	1 |- (A.x(x = A -> B = C) <-> [_A / x]_B = C)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  Vcvv 1811  [_csb 2001

This theorem is referenced by:  csbief 2032

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-10 966  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-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

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