Theorem csbiebg 3135

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 2214	. . . 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 1846	. 2 |- ( a = A -> ( A. x ( x = a -> B = C ) <-> A. x ( x = A -> B = C ) ) )
4		csbeq1 3095	. . 3 |- ( a = A -> [_ a / x ]_ B = [_ A / x ]_ B )
5	4	eqeq1d 2213	. 2 |- ( a = A -> ( [_ a / x ]_ B = C <-> [_ A / x ]_ B = C ) )
6		vex 2774	. . 3 |- a e. _V
7		csbiebg.2	. . 3 |- F/_ x C
8	6, 7	csbieb 3134	. 2 |- ( A. x ( x = a -> B = C ) <-> [_ a / x ]_ B = C )
9	3, 5, 8	vtoclbg 2833	1 |- ( A e. V -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1370   = wceq 1372   e. wcel 2175   F/_wnfc 2334   [_csb 3092

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-sbc 2998  df-csb 3093

This theorem is referenced by: (None)