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Theorem csbrecsg 37329
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 8346 . . 3 (𝐴𝑉𝐴 / 𝑥wrecs( E , On, 𝐹) = wrecs(𝐴 / 𝑥 E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹))
2 csbconstg 3918 . . . 4 (𝐴𝑉𝐴 / 𝑥 E = E )
3 wrecseq1 8343 . . . 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 3918 . . . 4 (𝐴𝑉𝐴 / 𝑥On = On)
6 wrecseq2 8344 . . . 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 2781 . 2 (𝐴𝑉𝐴 / 𝑥wrecs( E , On, 𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹))
9 df-recs 8411 . . 3 recs(𝐹) = wrecs( E , On, 𝐹)
109csbeq2i 3907 . 2 𝐴 / 𝑥recs(𝐹) = 𝐴 / 𝑥wrecs( E , On, 𝐹)
11 df-recs 8411 . 2 recs(𝐴 / 𝑥𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹)
128, 10, 113eqtr4g 2802 1 (𝐴𝑉𝐴 / 𝑥recs(𝐹) = recs(𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  csb 3899   E cep 5583  Oncon0 6384  wrecscwrecs 8336  recscrecs 8410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fv 6569  df-ov 7434  df-frecs 8306  df-wrecs 8337  df-recs 8411
This theorem is referenced by:  csbrdgg  37330
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