Theorem predeq3 6263

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 2736	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2736	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6260	. 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 6258

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 2115  ax-9 2123  ax-ext 2708

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 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259

This theorem is referenced by:  dfpred3g  6271  preddowncl  6290  frpoinsg  6301  frpoins3xpg  8082  frpoins3xp3g  8083  xpord2pred  8087  sexp2  8088  xpord3pred  8094  sexp3  8095  csbfrecsg  8226  fpr3g  8227  frrlem1  8228  frrlem12  8239  frrlem13  8240  fpr2a  8244  frrdmcl  8250  fprresex  8252  wfr3g  8261  ttrclselem1  9634  ttrclselem2  9635  frmin  9661  frinsg  9663  frr3g  9668  frr2  9672  elwlim  36015