Theorem predeq3 6303

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 2760	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2760	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6300	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1480	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Predcpred 6298

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299

This theorem is used by:  dfpred3g  6311  preddowncl  6330  frpoinsg  6341  frpoins3xpg  8138  frpoins3xp3g  8139  xpord2pred  8143  sexp2  8144  xpord3pred  8150  sexp3  8151  csbfrecsg  8283  fpr3g  8284  frrlem1  8285  frrlem12  8296  frrlem13  8297  fpr2a  8301  frrdmcl  8307  fprresex  8309  wfr3g  8318  ttrclselem1  9704  ttrclselem2  9705  frmin  9731  frinsg  9733  frr3g  9738  frr2  9742  elwlim  36400