Theorem cvbtrcl 13731

Description: Change of bound variable in class of all transitive relations which are supersets of a relation. (Contributed by RP, 5-May-2020.)

Assertion

Ref	Expression
cvbtrcl	|- { x | ( R C_ x /\ ( x o. x ) C_ x ) } = { y | ( R C_ y /\ ( y o. y ) C_ y ) }

Distinct variable group:   x, y, R

Proof of Theorem cvbtrcl

Step	Hyp	Ref	Expression
1		trcleq2lem 13730	. 2 |- ( x = y -> ( ( R C_ x /\ ( x o. x ) C_ x ) <-> ( R C_ y /\ ( y o. y ) C_ y ) ) )
2	1	cbvabv 2747	1 |- { x | ( R C_ x /\ ( x o. x ) C_ x ) } = { y | ( R C_ y /\ ( y o. y ) C_ y ) }

Syntax hints:   /\ wa 384   = wceq 1483   {cab 2608   C_ wss 3574   o. ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-co 5123

This theorem is referenced by: (None)