Theorem predeq3 6294

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 6291	. 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 1540  Predcpred 6289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290

This theorem is referenced by:  dfpred3g  6302  preddowncl  6321  frpoinsg  6332  wfisgOLD  6343  frpoins3xpg  8139  frpoins3xp3g  8140  xpord2pred  8144  sexp2  8145  xpord3pred  8151  sexp3  8152  csbfrecsg  8283  fpr3g  8284  frrlem1  8285  frrlem12  8296  frrlem13  8297  fpr2a  8301  frrdmcl  8307  fprresex  8309  wfr3g  8321  wfrlem1OLD  8322  wfrdmclOLD  8331  wfrlem14OLD  8336  wfrlem15OLD  8337  wfrlem17OLD  8339  wfr2aOLD  8340  ttrclselem1  9739  ttrclselem2  9740  frmin  9763  frinsg  9765  frr3g  9770  frr2  9774  elwlim  35841