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Theorem rdgeq12 7908
 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 7907 . 2 (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
2 rdgeq1 7906 . 2 (𝐹 = 𝐺 → rec(𝐹, 𝐵) = rec(𝐺, 𝐵))
31, 2sylan9eqr 2855 1 ((𝐹 = 𝐺𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1525  reccrdg 7904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-xp 5456  df-cnv 5458  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-iota 6196  df-fv 6240  df-wrecs 7805  df-recs 7867  df-rdg 7905 This theorem is referenced by:  seqomeq12  7948  seqeq3  13228  satf  32210  satf0  32229  trpredeq1  32670  trpredeq2  32671  trpred0  32686  csbfinxpg  34221
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