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Theorem csbrecsg 33504
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 33503 . . 3 (𝐴𝑉𝐴 / 𝑥wrecs( E , On, 𝐹) = wrecs(𝐴 / 𝑥 E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹))
2 csbconstg 3693 . . . 4 (𝐴𝑉𝐴 / 𝑥 E = E )
3 wrecseq1 7561 . . . 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 3693 . . . 4 (𝐴𝑉𝐴 / 𝑥On = On)
6 wrecseq2 7562 . . . 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 2808 . 2 (𝐴𝑉𝐴 / 𝑥wrecs( E , On, 𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹))
9 df-recs 7620 . . 3 recs(𝐹) = wrecs( E , On, 𝐹)
109csbeq2i 4135 . 2 𝐴 / 𝑥recs(𝐹) = 𝐴 / 𝑥wrecs( E , On, 𝐹)
11 df-recs 7620 . 2 recs(𝐴 / 𝑥𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹)
128, 10, 113eqtr4g 2829 1 (𝐴𝑉𝐴 / 𝑥recs(𝐹) = recs(𝐴 / 𝑥𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  csb 3680   E cep 5161  Oncon0 5866  wrecscwrecs 7557  recscrecs 7619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-iota 5994  df-fv 6039  df-wrecs 7558  df-recs 7620
This theorem is referenced by:  csbrdgg  33505
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