Theorem eqsbc2 3023

Description: Substitution for the right-hand side in an equality. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)

Assertion

Ref	Expression
eqsbc2	|- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) )

Distinct variable group:   x, B

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

Proof of Theorem eqsbc2

Step	Hyp	Ref	Expression
1		eqsbc1 3002	. 2 |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )
2		eqcom 2179	. . 3 |- ( B = x <-> x = B )
3	2	sbcbii 3022	. 2 |- ( [. A / x ]. B = x <-> [. A / x ]. x = B )
4		eqcom 2179	. 2 |- ( B = A <-> A = B )
5	1, 3, 4	3bitr4g 223	1 |- ( A e. V -> ( [. A / x ]. B = x <-> B = A ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   [.wsbc 2962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963

This theorem is referenced by: (None)