Theorem csbrecsg 33174

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

Assertion

Ref	Expression
csbrecsg	|- ( A e. V -> [_ A / x ]_recs ( F ) = recs ( [_ A / x ]_ F ) )

Proof of Theorem csbrecsg

Step	Hyp	Ref	Expression
1		csbwrecsg 33173	. . 3 |- ( A e. V -> [_ A / x ]_wrecs ( _E , On , F ) = wrecs ( [_ A / x ]_ _E , [_ A / x ]_ On , [_ A / x ]_ F ) )
2		csbconstg 3546	. . . 4 |- ( A e. V -> [_ A / x ]_ _E = _E )
3		wrecseq1 7410	. . . 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 e. V -> wrecs ( [_ A / x ]_ _E , [_ A / x ]_ On , [_ A / x ]_ F ) = wrecs ( _E , [_ A / x ]_ On , [_ A / x ]_ F ) )
5		csbconstg 3546	. . . 4 |- ( A e. V -> [_ A / x ]_ On = On )
6		wrecseq2 7411	. . . 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 e. V -> wrecs ( _E , [_ A / x ]_ On , [_ A / x ]_ F ) = wrecs ( _E , On , [_ A / x ]_ F ) )
8	1, 4, 7	3eqtrd 2660	. 2 |- ( A e. V -> [_ A / x ]_wrecs ( _E , On , F ) = wrecs ( _E , On , [_ A / x ]_ F ) )
9		df-recs 7468	. . 3 |- recs ( F ) = wrecs ( _E , On , F )
10	9	csbeq2i 3993	. 2 |- [_ A / x ]_recs ( F ) = [_ A / x ]_wrecs ( _E , On , F )
11		df-recs 7468	. 2 |- recs ( [_ A / x ]_ F ) = wrecs ( _E , On , [_ A / x ]_ F )
12	8, 10, 11	3eqtr4g 2681	1 |- ( A e. V -> [_ A / x ]_recs ( F ) = recs ( [_ A / x ]_ F ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   [_csb 3533   _E cep 5028   Oncon0 5723  wrecscwrecs 7406  recscrecs 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  csbrdgg  33175