Theorem predeq3 5984

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 2772	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2772	. 2 ⊢ 𝐴 = 𝐴
3		predeq123 5981	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1430	1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1507  Predcpred 5979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-xp 5406  df-cnv 5408  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980

This theorem is referenced by:  dfpred3g  5991  predbrg  6000  preddowncl  6007  wfisg  6015  wfr3g  7749  wfrlem1  7750  wfrdmcl  7760  wfrlem14  7765  wfrlem15  7766  wfrlem17  7768  wfr2a  7769  trpredeq3  32522  trpredlem1  32527  trpredtr  32530  trpredmintr  32531  trpredrec  32538  frpoinsg  32542  frmin  32545  frinsg  32548  elwlim  32571  frr3g  32582  fpr3g  32583  frrlem1  32584  frrlem12  32595  frrlem13  32596  fpr2  32601  frr2  32606  csbwrecsg  33985