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Theorem csbrecsg 32350
 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 32349 . . 3 (𝐴𝑉𝐴 / 𝑥wrecs( E , On, 𝐹) = wrecs(𝐴 / 𝑥 E , 𝐴 / 𝑥On, 𝐴 / 𝑥𝐹))
2 csbconstg 3512 . . . 4 (𝐴𝑉𝐴 / 𝑥 E = E )
3 wrecseq1 7297 . . . 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 3512 . . . 4 (𝐴𝑉𝐴 / 𝑥On = On)
6 wrecseq2 7298 . . . 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 2648 . 2 (𝐴𝑉𝐴 / 𝑥wrecs( E , On, 𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹))
9 df-recs 7355 . . 3 recs(𝐹) = wrecs( E , On, 𝐹)
109csbeq2i 3945 . 2 𝐴 / 𝑥recs(𝐹) = 𝐴 / 𝑥wrecs( E , On, 𝐹)
11 df-recs 7355 . 2 recs(𝐴 / 𝑥𝐹) = wrecs( E , On, 𝐴 / 𝑥𝐹)
128, 10, 113eqtr4g 2669 1 (𝐴𝑉𝐴 / 𝑥recs(𝐹) = recs(𝐴 / 𝑥𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⦋csb 3499   E cep 4947  Oncon0 5640  wrecscwrecs 7293  recscrecs 7354 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-wrecs 7294  df-recs 7355 This theorem is referenced by:  csbrdgg  32351
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