Theorem csbieb 3555

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 3553	. 2 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
4	1, 2, 3	mp2an 708	1 |- ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  csbiebg  3556