Theorem predeq3 6247

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 2731	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2731	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6244	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1453	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1541  Predcpred 6242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-xp 5617  df-cnv 5619  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243

This theorem is referenced by:  dfpred3g  6255  preddowncl  6274  frpoinsg  6285  frpoins3xpg  8065  frpoins3xp3g  8066  xpord2pred  8070  sexp2  8071  xpord3pred  8077  sexp3  8078  csbfrecsg  8209  fpr3g  8210  frrlem1  8211  frrlem12  8222  frrlem13  8223  fpr2a  8227  frrdmcl  8233  fprresex  8235  wfr3g  8244  ttrclselem1  9610  ttrclselem2  9611  frmin  9637  frinsg  9639  frr3g  9644  frr2  9648  elwlim  35857