Theorem csbieb 3036

Description: Bidirectional conversion between an implicit class substitution hypothesis x = A -> B = C and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)

Hypotheses

Ref	Expression
csbieb.1	|- A e. _V
csbieb.2	|- F/_ x C

Assertion

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

Distinct variable group:   x, A

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

Proof of Theorem csbieb

Step	Hyp	Ref	Expression
1		csbieb.1	. 2 |- A e. _V
2		csbieb.2	. 2 |- F/_ x C
3		csbiebt 3034	. 2 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
4	1, 2, 3	mp2an 422	1 |- ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   e. wcel 1480   F/_wnfc 2266   _Vcvv 2681   [_csb 2998

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999

This theorem is referenced by:  csbiebg  3037