Theorem predeq3 6195

Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Assertion

Ref	Expression
predeq3	⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Proof of Theorem predeq3

Step	Hyp	Ref	Expression
1		eqid 2738	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2738	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6192	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1449	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1539  Predcpred 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191

This theorem is referenced by:  dfpred3g  6203  predbrg  6213  preddowncl  6224  frpoinsg  6231  wfisgOLD  6242  csbfrecsg  8071  fpr3g  8072  frrlem1  8073  frrlem12  8084  frrlem13  8085  fpr2a  8089  frrdmcl  8095  fprresex  8097  wfr3g  8109  wfrlem1OLD  8110  wfrdmclOLD  8119  wfrlem14OLD  8124  wfrlem15OLD  8125  wfrlem17OLD  8127  wfr2aOLD  8128  trpredeq3  9400  trpredlem1  9405  trpredtr  9408  trpredmintr  9409  trpredrec  9415  frmin  9438  frinsg  9440  frr3g  9445  frr2  9449  ttrclselem1  33711  ttrclselem2  33712  frpoins3xpg  33714  frpoins3xp3g  33715  xpord2pred  33719  sexp2  33720  xpord3pred  33725  sexp3  33726  elwlim  33744