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

Theorem rdgeq12 8408
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 8407 . 2 (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
2 rdgeq1 8406 . 2 (𝐹 = 𝐺 → rec(𝐹, 𝐵) = rec(𝐺, 𝐵))
31, 2sylan9eqr 2795 1 ((𝐹 = 𝐺𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  reccrdg 8404
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-xp 5681  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-iota 6492  df-fv 6548  df-ov 7407  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405
This theorem is referenced by:  seqomeq12  8449  seqeq3  13967  satf  34282  satf0  34301  csbfinxpg  36207
  Copyright terms: Public domain W3C validator