Theorem predeq3 6327

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 2735	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2735	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6324	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1450	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1537  Predcpred 6322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323

This theorem is referenced by:  dfpred3g  6335  preddowncl  6355  frpoinsg  6366  wfisgOLD  6377  frpoins3xpg  8164  frpoins3xp3g  8165  xpord2pred  8169  sexp2  8170  xpord3pred  8176  sexp3  8177  csbfrecsg  8308  fpr3g  8309  frrlem1  8310  frrlem12  8321  frrlem13  8322  fpr2a  8326  frrdmcl  8332  fprresex  8334  wfr3g  8346  wfrlem1OLD  8347  wfrdmclOLD  8356  wfrlem14OLD  8361  wfrlem15OLD  8362  wfrlem17OLD  8364  wfr2aOLD  8365  ttrclselem1  9763  ttrclselem2  9764  frmin  9787  frinsg  9789  frr3g  9794  frr2  9798  elwlim  35805