Theorem bnj602 29722

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 4424	. . 3 |- ( X = Y -> ( y R X <-> y R Y ) )
2	1	rabbidv 3072	. 2 |- ( X = Y -> { y e. A | y R X } = { y e. A | y R Y } )
3		df-bnj14 29490	. 2 |- pred ( X , A , R ) = { y e. A | y R X }
4		df-bnj14 29490	. 2 |- pred ( Y , A , R ) = { y e. A | y R Y }
5	2, 3, 4	3eqtr4g 2488	1 |- ( X = Y -> pred ( X , A , R ) = pred ( Y , A , R ) )

Syntax hints:   -> wi 4   = wceq 1437   {crab 2779   class class class wbr 4420   predc-bnj14 29489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-bnj14 29490

This theorem is referenced by:  bnj601  29727  bnj852  29728  bnj18eq1  29734  bnj938  29744  bnj1125  29797  bnj1148  29801  bnj1318  29830  bnj1442  29854  bnj1450  29855  bnj1452  29857  bnj1463  29860  bnj1529  29875