MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rdgeq12 Structured version   Visualization version   GIF version

Theorem rdgeq12 7368
Description: Equality theorem for the recursive definition generator. (Contributed by Scott Fenton, 28-Apr-2012.)
Assertion
Ref Expression
rdgeq12 ((𝐹 = 𝐺𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))

Proof of Theorem rdgeq12
StepHypRef Expression
1 rdgeq2 7367 . 2 (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
2 rdgeq1 7366 . 2 (𝐹 = 𝐺 → rec(𝐹, 𝐵) = rec(𝐺, 𝐵))
31, 2sylan9eqr 2660 1 ((𝐹 = 𝐺𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  reccrdg 7364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-mpt 4634  df-xp 5029  df-cnv 5031  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-iota 5749  df-fv 5793  df-wrecs 7266  df-recs 7327  df-rdg 7365
This theorem is referenced by:  seqomeq12  7408  seqeq3  12618  trpredeq1  30765  trpredeq2  30766  trpred0  30781  csbfinxpg  32199
  Copyright terms: Public domain W3C validator