Theorem csbiebg 3556

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 2633	. . . 4 |- ( a = A -> ( x = a <-> x = A ) )
2	1	imbi1d 331	. . 3 |- ( a = A -> ( ( x = a -> B = C ) <-> ( x = A -> B = C ) ) )
3	2	albidv 1849	. 2 |- ( a = A -> ( A. x ( x = a -> B = C ) <-> A. x ( x = A -> B = C ) ) )
4		csbeq1 3536	. . 3 |- ( a = A -> [_ a / x ]_ B = [_ A / x ]_ B )
5	4	eqeq1d 2624	. 2 |- ( a = A -> ( [_ a / x ]_ B = C <-> [_ A / x ]_ B = C ) )
6		vex 3203	. . 3 |- a e. _V
7		csbiebg.2	. . 3 |- F/_ x C
8	6, 7	csbieb 3555	. 2 |- ( A. x ( x = a -> B = C ) <-> [_ a / x ]_ B = C )
9	3, 5, 8	vtoclbg 3267	1 |- ( A e. V -> ( 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   [_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:  cdlemefrs29bpre0  35684