Theorem predeq3 6307

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 2769	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2769	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6304	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1477	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1567  Predcpred 6302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303

This theorem is referenced by:  dfpred3g  6315  preddowncl  6334  frpoinsg  6345  frpoins3xpg  8135  frpoins3xp3g  8136  xpord2pred  8140  sexp2  8141  xpord3pred  8147  sexp3  8148  csbfrecsg  8280  fpr3g  8281  frrlem1  8282  frrlem12  8293  frrlem13  8294  fpr2a  8298  frrdmcl  8304  fprresex  8306  wfr3g  8315  ttrclselem1  9693  ttrclselem2  9694  frmin  9720  frinsg  9722  frr3g  9727  frr2  9731  elwlim  36211