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Theorem csbrecsg 34608
Description: Move class substitution in and out of recs. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csbrecsg (𝐴𝑉𝐴 / 𝑥recs(𝐹) = recs(𝐴 / 𝑥𝐹))

Proof of Theorem csbrecsg
StepHypRef Expression
1 csbwrecsg 34607 . . 3 (𝐴𝑉𝐴 / 𝑥wrecs( E , On, 𝐹) = wrecs(𝐴 / 𝑥 E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹))
2 csbconstg 3901 . . . 4 (𝐴𝑉𝐴 / 𝑥 E = E )
3 wrecseq1 7949 . . . 4 (𝐴 / 𝑥 E = E → wrecs(𝐴 / 𝑥 E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹) = wrecs( E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹))
42, 3syl 17 . . 3 (𝐴𝑉 → wrecs(𝐴 / 𝑥 E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹) = wrecs( E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹))
5 csbconstg 3901 . . . 4 (𝐴𝑉𝐴 / 𝑥On = On)
6 wrecseq2 7950 . . . 4 (𝐴 / 𝑥On = On → wrecs( E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹))
75, 6syl 17 . . 3 (𝐴𝑉 → wrecs( E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹))
81, 4, 73eqtrd 2860 . 2 (𝐴𝑉𝐴 / 𝑥wrecs( E , On, 𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹))
9 df-recs 8007 . . 3 recs(𝐹) = wrecs( E , On, 𝐹)
109csbeq2i 3890 . 2 𝐴 / 𝑥recs(𝐹) = 𝐴 / 𝑥wrecs( E , On, 𝐹)
11 df-recs 8007 . 2 recs(𝐴 / 𝑥𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹)
128, 10, 113eqtr4g 2881 1 (𝐴𝑉𝐴 / 𝑥recs(𝐹) = recs(𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  csb 3882   E cep 5463  Oncon0 6190  wrecscwrecs 7945  recscrecs 8006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-xp 5560  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-iota 6313  df-fv 6362  df-wrecs 7946  df-recs 8007
This theorem is referenced by:  csbrdgg  34609
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