Theorem predeq3 6257

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 2729	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2729	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6254	. 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 6252

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253

This theorem is referenced by:  dfpred3g  6265  preddowncl  6284  frpoinsg  6295  frpoins3xpg  8080  frpoins3xp3g  8081  xpord2pred  8085  sexp2  8086  xpord3pred  8092  sexp3  8093  csbfrecsg  8224  fpr3g  8225  frrlem1  8226  frrlem12  8237  frrlem13  8238  fpr2a  8242  frrdmcl  8248  fprresex  8250  wfr3g  8259  ttrclselem1  9640  ttrclselem2  9641  frmin  9664  frinsg  9666  frr3g  9671  frr2  9675  elwlim  35816