Theorem predeq3 6271

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 2737	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2737	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6268	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1454	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1542  Predcpred 6266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267

This theorem is referenced by:  dfpred3g  6279  preddowncl  6298  frpoinsg  6309  frpoins3xpg  8092  frpoins3xp3g  8093  xpord2pred  8097  sexp2  8098  xpord3pred  8104  sexp3  8105  csbfrecsg  8236  fpr3g  8237  frrlem1  8238  frrlem12  8249  frrlem13  8250  fpr2a  8254  frrdmcl  8260  fprresex  8262  wfr3g  8271  ttrclselem1  9646  ttrclselem2  9647  frmin  9673  frinsg  9675  frr3g  9680  frr2  9684  elwlim  36037