Theorem bnj602 33408

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 4457	. . 3 |- ( X = Y -> ( y R X <-> y R Y ) )
2	1	rabbidv 3110	. 2 |- ( X = Y -> { y e. A | y R X } = { y e. A | y R Y } )
3		df-bnj14 33177	. 2 |- pred ( X , A , R ) = { y e. A | y R X }
4		df-bnj14 33177	. 2 |- pred ( Y , A , R ) = { y e. A | y R Y }
5	2, 3, 4	3eqtr4g 2533	1 |- ( X = Y -> pred ( X , A , R ) = pred ( Y , A , R ) )

Syntax hints:   -> wi 4   = wceq 1379   {crab 2821   class class class wbr 4453   predc-bnj14 33176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-bnj14 33177

This theorem is referenced by:  bnj601  33413  bnj852  33414  bnj18eq1  33420  bnj938  33430  bnj1125  33483  bnj1148  33487  bnj1318  33516  bnj1442  33540  bnj1450  33541  bnj1452  33543  bnj1463  33546  bnj1529  33561