Theorem predeq3 6264

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 2737	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2737	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6261	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1454	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1542  Predcpred 6259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260

This theorem is referenced by:  dfpred3g  6272  preddowncl  6291  frpoinsg  6302  frpoins3xpg  8084  frpoins3xp3g  8085  xpord2pred  8089  sexp2  8090  xpord3pred  8096  sexp3  8097  csbfrecsg  8228  fpr3g  8229  frrlem1  8230  frrlem12  8241  frrlem13  8242  fpr2a  8246  frrdmcl  8252  fprresex  8254  wfr3g  8263  ttrclselem1  9640  ttrclselem2  9641  frmin  9667  frinsg  9669  frr3g  9674  frr2  9678  elwlim  36022