Theorem predeq3 6334

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 6331	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1481	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Predcpred 6329

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330

This theorem is used by:  dfpred3g  6342  preddowncl  6361  frpoinsg  6372  frpoins3xpg  8194  frpoins3xp3g  8195  xpord2pred  8199  sexp2  8200  xpord3pred  8206  sexp3  8207  csbfrecsg  8340  fpr3g  8341  frrlem1  8342  frrlem12  8353  frrlem13  8354  fpr2a  8358  frrdmcl  8364  fprresex  8366  wfr3g  8375  ttrclselem1  9763  ttrclselem2  9764  frmin  9790  frinsg  9792  frr3g  9797  frr2  9801  elwlim  36800