Theorem csbieb 3189

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 3187	. 2 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
4	1, 2, 3	mp2an 430	1 |- ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1400   = wceq 1402   e. wcel 2209   F/_wnfc 2379   _Vcvv 2821   [_csb 3147

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-csb 3148

This theorem is referenced by:  csbiebg  3190