Theorem eqsbc2 3015

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 2994	. 2 |- ( A e. V -> ( [. A / x ]. x = B <-> A = B ) )
2		eqcom 2172	. . 3 |- ( B = x <-> x = B )
3	2	sbcbii 3014	. 2 |- ( [. A / x ]. B = x <-> [. A / x ]. x = B )
4		eqcom 2172	. 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 1348   e. wcel 2141   [.wsbc 2955

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956

This theorem is referenced by: (None)