Theorem predeq3 6310

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 2765	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2765	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6307	. 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))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Predcpred 6305

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306

This theorem is used by:  dfpred3g  6318  preddowncl  6337  frpoinsg  6348  frpoins3xpg  8142  frpoins3xp3g  8143  xpord2pred  8147  sexp2  8148  xpord3pred  8154  sexp3  8155  csbfrecsg  8287  fpr3g  8288  frrlem1  8289  frrlem12  8300  frrlem13  8301  fpr2a  8305  frrdmcl  8311  fprresex  8313  wfr3g  8322  ttrclselem1  9701  ttrclselem2  9702  frmin  9728  frinsg  9730  frr3g  9735  frr2  9739  elwlim  36350