Theorem csbieb 3170

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 3168	. 2 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
4	1, 2, 3	mp2an 426	1 |- ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1396   = wceq 1398   e. wcel 2202   F/_wnfc 2362   _Vcvv 2803   [_csb 3128

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sbc 3033  df-csb 3129

This theorem is referenced by:  csbiebg  3171