Theorem bnj602 32221

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 4399	. . 3 |- ( X = Y -> ( y R X <-> y R Y ) )
2	1	rabbidv 3064	. 2 |- ( X = Y -> { y e. A | y R X } = { y e. A | y R Y } )
3		df-bnj14 31990	. 2 |- pred ( X , A , R ) = { y e. A | y R X }
4		df-bnj14 31990	. 2 |- pred ( Y , A , R ) = { y e. A | y R Y }
5	2, 3, 4	3eqtr4g 2518	1 |- ( X = Y -> pred ( X , A , R ) = pred ( Y , A , R ) )

Syntax hints:   -> wi 4   = wceq 1370   {crab 2800   class class class wbr 4395   predc-bnj14 31989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-bnj14 31990

This theorem is referenced by:  bnj601  32226  bnj852  32227  bnj18eq1  32233  bnj938  32243  bnj1125  32296  bnj1148  32300  bnj1318  32329  bnj1442  32353  bnj1450  32354  bnj1452  32356  bnj1463  32359  bnj1529  32374