Theorem csbiebg 3097

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 2185	. . . 4 |- ( a = A -> ( x = a <-> x = A ) )
2	1	imbi1d 231	. . 3 |- ( a = A -> ( ( x = a -> B = C ) <-> ( x = A -> B = C ) ) )
3	2	albidv 1822	. 2 |- ( a = A -> ( A. x ( x = a -> B = C ) <-> A. x ( x = A -> B = C ) ) )
4		csbeq1 3058	. . 3 |- ( a = A -> [_ a / x ]_ B = [_ A / x ]_ B )
5	4	eqeq1d 2184	. 2 |- ( a = A -> ( [_ a / x ]_ B = C <-> [_ A / x ]_ B = C ) )
6		vex 2738	. . 3 |- a e. _V
7		csbiebg.2	. . 3 |- F/_ x C
8	6, 7	csbieb 3096	. 2 |- ( A. x ( x = a -> B = C ) <-> [_ a / x ]_ B = C )
9	3, 5, 8	vtoclbg 2796	1 |- ( A e. V -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2146   F/_wnfc 2304   [_csb 3055

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-sbc 2961  df-csb 3056

This theorem is referenced by: (None)