Theorem csbiebg 3091

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 2180	. . . 4 |- ( a = A -> ( x = a <-> x = A ) )
2	1	imbi1d 230	. . 3 |- ( a = A -> ( ( x = a -> B = C ) <-> ( x = A -> B = C ) ) )
3	2	albidv 1817	. 2 |- ( a = A -> ( A. x ( x = a -> B = C ) <-> A. x ( x = A -> B = C ) ) )
4		csbeq1 3052	. . 3 |- ( a = A -> [_ a / x ]_ B = [_ A / x ]_ B )
5	4	eqeq1d 2179	. 2 |- ( a = A -> ( [_ a / x ]_ B = C <-> [_ A / x ]_ B = C ) )
6		vex 2733	. . 3 |- a e. _V
7		csbiebg.2	. . 3 |- F/_ x C
8	6, 7	csbieb 3090	. 2 |- ( A. x ( x = a -> B = C ) <-> [_ a / x ]_ B = C )
9	3, 5, 8	vtoclbg 2791	1 |- ( A e. V -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141   F/_wnfc 2299   [_csb 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by: (None)