Theorem bnj602 33841

Description: Equality theorem for the pred function constant. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj602	|- ( X = Y -> pred ( X , A , R ) = pred ( Y , A , R ) )

Proof of Theorem bnj602

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4441	. . 3 |- ( X = Y -> ( y R X <-> y R Y ) )
2	1	rabbidv 3087	. 2 |- ( X = Y -> { y e. A | y R X } = { y e. A | y R Y } )
3		df-bnj14 33609	. 2 |- pred ( X , A , R ) = { y e. A | y R X }
4		df-bnj14 33609	. 2 |- pred ( Y , A , R ) = { y e. A | y R Y }
5	2, 3, 4	3eqtr4g 2509	1 |- ( X = Y -> pred ( X , A , R ) = pred ( Y , A , R ) )

Syntax hints:   -> wi 4   = wceq 1383   {crab 2797   class class class wbr 4437   predc-bnj14 33608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-bnj14 33609

This theorem is referenced by:  bnj601  33846  bnj852  33847  bnj18eq1  33853  bnj938  33863  bnj1125  33916  bnj1148  33920  bnj1318  33949  bnj1442  33973  bnj1450  33974  bnj1452  33976  bnj1463  33979  bnj1529  33994