Theorem eqsbc2 3046

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 3025	. 2 |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )
2		eqcom 2195	. . 3 |- ( B = x <-> x = B )
3	2	sbcbii 3045	. 2 |- ( [. A / x ]. B = x <-> [. A / x ]. x = B )
4		eqcom 2195	. 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 1364   e. wcel 2164   [.wsbc 2985

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986

This theorem is referenced by: (None)