Theorem csbov2g 7438

Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)

Assertion

Ref	Expression
csbov2g	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵𝐹⦋𝐴 / 𝑥⦌𝐶))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹

Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbov2g

Step	Hyp	Ref	Expression
1		csbov12g 7436	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶))
2		csbconstg 3869	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
3	2	oveq1d 7405	. 2 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵𝐹⦋𝐴 / 𝑥⦌𝐶))
4	1, 3	eqtrd 2796	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵𝐹⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ⦋csb 3850  (class class class)co 7390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-dm 5653  df-iota 6471  df-fv 6523  df-ov 7393

This theorem is referenced by:  csbnegg  11420  prmgaplem7  17083  matgsum  22484  scmatscm  22560  pm2mpf1lem  22841  pm2mpcoe1  22847  pm2mpmhmlem2  22866  monmat2matmon  22871  divcncf  25496  logbmpt  26840  finxpreclem4  37848  tfsconcatfv  43878  cotrclrcl  44278  ply1mulgsumlem3  48970  ply1mulgsumlem4  48971  ply1mulgsum  48972