Theorem eqsbc3r 2964

Description: eqsbc3 2943 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 2943	. 2 |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )
2		eqcom 2139	. . 3 |- ( B = x <-> x = B )
3	2	sbcbii 2963	. 2 |- ( [. A / x ]. B = x <-> [. A / x ]. x = B )
4		eqcom 2139	. 2 |- ( B = A <-> A = B )
5	1, 3, 4	3bitr4g 222	1 |- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   [.wsbc 2904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905

This theorem is referenced by: (None)