Theorem predeq3 6292

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 2762	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2762	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6289	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1472	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1560  Predcpred 6287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288

This theorem is referenced by:  dfpred3g  6300  preddowncl  6319  frpoinsg  6330  frpoins3xpg  8120  frpoins3xp3g  8121  xpord2pred  8125  sexp2  8126  xpord3pred  8132  sexp3  8133  csbfrecsg  8265  fpr3g  8266  frrlem1  8267  frrlem12  8278  frrlem13  8279  fpr2a  8283  frrdmcl  8289  fprresex  8291  wfr3g  8300  ttrclselem1  9680  ttrclselem2  9681  frmin  9707  frinsg  9709  frr3g  9714  frr2  9718  elwlim  36171