Theorem eqsbc3r 2899

Description: eqsbc3 2878 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)

Assertion

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

Distinct variable group:   x, B

Allowed substitution hints:   A( x)   V( x)

Proof of Theorem eqsbc3r

Step	Hyp	Ref	Expression
1		eqsbc3 2878	. 2 |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )
2		eqcom 2090	. . 3 |- ( B = x <-> x = B )
3	2	sbcbii 2898	. 2 |- ( [. A / x ]. B = x <-> [. A / x ]. x = B )
4		eqcom 2090	. 2 |- ( B = A <-> A = B )
5	1, 3, 4	3bitr4g 221	1 |- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   e. wcel 1438   [.wsbc 2840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841

This theorem is referenced by: (None)