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Theorem rdgeq12 8059
 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 8058 . 2 (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
2 rdgeq1 8057 . 2 (𝐹 = 𝐺 → rec(𝐹, 𝐵) = rec(𝐺, 𝐵))
31, 2sylan9eqr 2815 1 ((𝐹 = 𝐺𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  reccrdg 8055 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rab 3079  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-xp 5530  df-cnv 5532  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-iota 6294  df-fv 6343  df-wrecs 7957  df-recs 8018  df-rdg 8056 This theorem is referenced by:  seqomeq12  8100  seqeq3  13423  satf  32831  satf0  32850  trpredeq1  33306  trpredeq2  33307  trpred0  33322  csbfinxpg  35085
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