Theorem csbrecsg 34745

Description: Move class substitution in and out of recs. (Contributed by ML, 25-Oct-2020.)

Assertion

Ref	Expression
csbrecsg	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌recs(𝐹) = recs(⦋𝐴 / 𝑥⦌𝐹))

Proof of Theorem csbrecsg

Step	Hyp	Ref	Expression
1		csbwrecsg 34744	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs( E , On, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌ E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹))
2		csbconstg 3847	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ E = E )
3		wrecseq1 7933	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌ E = E → wrecs(⦋𝐴 / 𝑥⦌ E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹) = wrecs( E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹))
4	2, 3	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → wrecs(⦋𝐴 / 𝑥⦌ E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹) = wrecs( E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹))
5		csbconstg 3847	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌On = On)
6		wrecseq2 7934	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌On = On → wrecs( E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹) = wrecs( E , On, ⦋𝐴 / 𝑥⦌𝐹))
7	5, 6	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → wrecs( E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹) = wrecs( E , On, ⦋𝐴 / 𝑥⦌𝐹))
8	1, 4, 7	3eqtrd 2837	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs( E , On, 𝐹) = wrecs( E , On, ⦋𝐴 / 𝑥⦌𝐹))
9		df-recs 7991	. . 3 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
10	9	csbeq2i 3836	. 2 ⊢ ⦋𝐴 / 𝑥⦌recs(𝐹) = ⦋𝐴 / 𝑥⦌wrecs( E , On, 𝐹)
11		df-recs 7991	. 2 ⊢ recs(⦋𝐴 / 𝑥⦌𝐹) = wrecs( E , On, ⦋𝐴 / 𝑥⦌𝐹)
12	8, 10, 11	3eqtr4g 2858	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌recs(𝐹) = recs(⦋𝐴 / 𝑥⦌𝐹))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ⦋csb 3828   E cep 5429  Oncon0 6159  wrecscwrecs 7929  recscrecs 7990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-iota 6283  df-fv 6332  df-wrecs 7930  df-recs 7991

This theorem is referenced by:  csbrdgg  34746