Theorem predeq3 6336

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 2740	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2740	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6333	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1451	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1537  Predcpred 6331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332

This theorem is referenced by:  dfpred3g  6344  preddowncl  6364  frpoinsg  6375  wfisgOLD  6386  frpoins3xpg  8181  frpoins3xp3g  8182  xpord2pred  8186  sexp2  8187  xpord3pred  8193  sexp3  8194  csbfrecsg  8325  fpr3g  8326  frrlem1  8327  frrlem12  8338  frrlem13  8339  fpr2a  8343  frrdmcl  8349  fprresex  8351  wfr3g  8363  wfrlem1OLD  8364  wfrdmclOLD  8373  wfrlem14OLD  8378  wfrlem15OLD  8379  wfrlem17OLD  8381  wfr2aOLD  8382  ttrclselem1  9794  ttrclselem2  9795  frmin  9818  frinsg  9820  frr3g  9825  frr2  9829  elwlim  35787