Theorem predeq3 6206

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 2738	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2738	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6203	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1450	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1539  Predcpred 6201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202

This theorem is referenced by:  dfpred3g  6214  predbrg  6224  preddowncl  6235  frpoinsg  6246  wfisgOLD  6257  csbfrecsg  8100  fpr3g  8101  frrlem1  8102  frrlem12  8113  frrlem13  8114  fpr2a  8118  frrdmcl  8124  fprresex  8126  wfr3g  8138  wfrlem1OLD  8139  wfrdmclOLD  8148  wfrlem14OLD  8153  wfrlem15OLD  8154  wfrlem17OLD  8156  wfr2aOLD  8157  ttrclselem1  9483  ttrclselem2  9484  frmin  9507  frinsg  9509  frr3g  9514  frr2  9518  frpoins3xpg  33787  frpoins3xp3g  33788  xpord2pred  33792  sexp2  33793  xpord3pred  33798  sexp3  33799  elwlim  33817