Theorem eqsbc3r 3291

Description: eqsbc3 3274 with setvar variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 37232 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 2458	. . . . . 6 |- ( B = x <-> x = B )
2	1	sbcbii 3290	. . . . 5 |- ( [. A / x ]. B = x <-> [. A / x ]. x = B )
3	2	biimpi 199	. . . 4 |- ( [. A / x ]. B = x -> [. A / x ]. x = B )
4		eqsbc3 3274	. . . 4 |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )
5	3, 4	syl5ib 227	. . 3 |- ( A e. V -> ( [. A / x ]. B = x -> A = B ) )
6		eqcom 2458	. . 3 |- ( A = B <-> B = A )
7	5, 6	syl6ib 234	. 2 |- ( A e. V -> ( [. A / x ]. B = x -> B = A ) )
8		idd 25	. . . . 5 |- ( A e. V -> ( B = A -> B = A ) )
9	8, 6	syl6ibr 235	. . . 4 |- ( A e. V -> ( B = A -> A = B ) )
10	9, 4	sylibrd 242	. . 3 |- ( A e. V -> ( B = A -> [. A / x ]. x = B ) )
11	10, 2	syl6ibr 235	. 2 |- ( A e. V -> ( B = A -> [. A / x ]. B = x ) )
12	7, 11	impbid 195	1 |- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1447   e. wcel 1890   [.wsbc 3234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3014  df-sbc 3235

This theorem is referenced by:  sbcoreleleq  36896  sbcoreleleqVD  37252