Theorem predeq3 6306

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 2763	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2763	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6303	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1480	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1570  Predcpred 6301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302

This theorem is referenced by:  dfpred3g  6314  preddowncl  6333  frpoinsg  6344  frpoins3xpg  8132  frpoins3xp3g  8133  xpord2pred  8137  sexp2  8138  xpord3pred  8144  sexp3  8145  csbfrecsg  8277  fpr3g  8278  frrlem1  8279  frrlem12  8290  frrlem13  8291  fpr2a  8295  frrdmcl  8301  fprresex  8303  wfr3g  8312  ttrclselem1  9690  ttrclselem2  9691  frmin  9717  frinsg  9719  frr3g  9724  frr2  9728  elwlim  36313