Theorem predeq3 6305

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 2733	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2733	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6302	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1452	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1542  Predcpred 6300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301

This theorem is referenced by:  dfpred3g  6313  predbrg  6323  preddowncl  6334  frpoinsg  6345  wfisgOLD  6356  frpoins3xpg  8126  frpoins3xp3g  8127  xpord2pred  8131  sexp2  8132  xpord3pred  8138  sexp3  8139  csbfrecsg  8269  fpr3g  8270  frrlem1  8271  frrlem12  8282  frrlem13  8283  fpr2a  8287  frrdmcl  8293  fprresex  8295  wfr3g  8307  wfrlem1OLD  8308  wfrdmclOLD  8317  wfrlem14OLD  8322  wfrlem15OLD  8323  wfrlem17OLD  8325  wfr2aOLD  8326  ttrclselem1  9720  ttrclselem2  9721  frmin  9744  frinsg  9746  frr3g  9751  frr2  9755  elwlim  34795