Theorem eqsbc3r 3243

Description: eqsbc3 3221 with setvar variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 31463 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
eqsbc3r	|- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem eqsbc3r

Step	Hyp	Ref	Expression
1		eqcom 2440	. . . . . 6 |- ( B = x <-> x = B )
2	1	sbcbii 3241	. . . . 5 |- ( [. A / x ]. B = x <-> [. A / x ]. x = B )
3	2	biimpi 194	. . . 4 |- ( [. A / x ]. B = x -> [. A / x ]. x = B )
4		eqsbc3 3221	. . . 4 |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )
5	3, 4	syl5ib 219	. . 3 |- ( A e. V -> ( [. A / x ]. B = x -> A = B ) )
6		eqcom 2440	. . 3 |- ( A = B <-> B = A )
7	5, 6	syl6ib 226	. 2 |- ( A e. V -> ( [. A / x ]. B = x -> B = A ) )
8		idd 24	. . . . 5 |- ( A e. V -> ( B = A -> B = A ) )
9	8, 6	syl6ibr 227	. . . 4 |- ( A e. V -> ( B = A -> A = B ) )
10	9, 4	sylibrd 234	. . 3 |- ( A e. V -> ( B = A -> [. A / x ]. x = B ) )
11	10, 2	syl6ibr 227	. 2 |- ( A e. V -> ( B = A -> [. A / x ]. B = x ) )
12	7, 11	impbid 191	1 |- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1369   e. wcel 1756   [.wsbc 3181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2969  df-sbc 3182

This theorem is referenced by:  sbcoreleleq  31128  sbcoreleleqVD  31482