Theorem csbiebg 3144

Description: Bidirectional conversion between an implicit class substitution hypothesis x = A -> B = C and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
csbiebg.2	|- F/_ x C

Assertion

Ref	Expression
csbiebg	|- ( A e. V -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   C( x)   V( x)

Proof of Theorem csbiebg

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2217	. . . 4 |- ( a = A -> ( x = a <-> x = A ) )
2	1	imbi1d 231	. . 3 |- ( a = A -> ( ( x = a -> B = C ) <-> ( x = A -> B = C ) ) )
3	2	albidv 1848	. 2 |- ( a = A -> ( A. x ( x = a -> B = C ) <-> A. x ( x = A -> B = C ) ) )
4		csbeq1 3104	. . 3 |- ( a = A -> [_ a / x ]_ B = [_ A / x ]_ B )
5	4	eqeq1d 2216	. 2 |- ( a = A -> ( [_ a / x ]_ B = C <-> [_ A / x ]_ B = C ) )
6		vex 2779	. . 3 |- a e. _V
7		csbiebg.2	. . 3 |- F/_ x C
8	6, 7	csbieb 3143	. 2 |- ( A. x ( x = a -> B = C ) <-> [_ a / x ]_ B = C )
9	3, 5, 8	vtoclbg 2839	1 |- ( A e. V -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2178   F/_wnfc 2337   [_csb 3101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sbc 3006  df-csb 3102

This theorem is referenced by: (None)