Theorem predeq3 6152

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 2821	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2821	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 6149	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1447	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1537  Predcpred 6147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148

This theorem is referenced by:  dfpred3g  6159  predbrg  6168  preddowncl  6175  wfisg  6183  wfr3g  7953  wfrlem1  7954  wfrdmcl  7963  wfrlem14  7968  wfrlem15  7969  wfrlem17  7971  wfr2a  7972  trpredeq3  33061  trpredlem1  33066  trpredtr  33069  trpredmintr  33070  trpredrec  33077  frpoinsg  33081  frmin  33084  frinsg  33087  elwlim  33110  frr3g  33121  fpr3g  33122  frrlem1  33123  frrlem12  33134  frrlem13  33135  fpr2  33140  frr2  33145  csbwrecsg  34611