Theorem eqsbc3r 2819

Description: eqsbc3 2802 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

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

Distinct variable groups:   x, C   x, A

Allowed substitution hint:   B( x)

Proof of Theorem eqsbc3r

Step	Hyp	Ref	Expression
1		eqcom 2042	. . . . . 6 |- ( C = x <-> x = C )
2	1	sbcbii 2818	. . . . 5 |- ( [. A / x ]. C = x <-> [. A / x ]. x = C )
3	2	biimpi 113	. . . 4 |- ( [. A / x ]. C = x -> [. A / x ]. x = C )
4		eqsbc3 2802	. . . 4 |- ( A e. B -> ( [. A / x ]. x = C <-> A = C ) )
5	3, 4	syl5ib 143	. . 3 |- ( A e. B -> ( [. A / x ]. C = x -> A = C ) )
6		eqcom 2042	. . 3 |- ( A = C <-> C = A )
7	5, 6	syl6ib 150	. 2 |- ( A e. B -> ( [. A / x ]. C = x -> C = A ) )
8		idd 21	. . . . 5 |- ( A e. B -> ( C = A -> C = A ) )
9	8, 6	syl6ibr 151	. . . 4 |- ( A e. B -> ( C = A -> A = C ) )
10	9, 4	sylibrd 158	. . 3 |- ( A e. B -> ( C = A -> [. A / x ]. x = C ) )
11	10, 2	syl6ibr 151	. 2 |- ( A e. B -> ( C = A -> [. A / x ]. C = x ) )
12	7, 11	impbid 120	1 |- ( A e. B -> ( [. A / x ]. C = x <-> C = A ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765

This theorem is referenced by: (None)