Theorem bnj602 31742

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 4293	. . 3 |- ( X = Y -> ( y R X <-> y R Y ) )
2	1	rabbidv 2962	. 2 |- ( X = Y -> { y e. A | y R X } = { y e. A | y R Y } )
3		df-bnj14 31511	. 2 |- pred ( X , A , R ) = { y e. A | y R X }
4		df-bnj14 31511	. 2 |- pred ( Y , A , R ) = { y e. A | y R Y }
5	2, 3, 4	3eqtr4g 2498	1 |- ( X = Y -> pred ( X , A , R ) = pred ( Y , A , R ) )

Syntax hints:   -> wi 4   = wceq 1364   {crab 2717   class class class wbr 4289   predc-bnj14 31510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-bnj14 31511

This theorem is referenced by:  bnj601  31747  bnj852  31748  bnj18eq1  31754  bnj938  31764  bnj1125  31817  bnj1148  31821  bnj1318  31850  bnj1442  31874  bnj1450  31875  bnj1452  31877  bnj1463  31880  bnj1529  31895