Theorem csbprg 4662

Description: Distribute proper substitution through a pair of classes. (Contributed by Alexander van der Vekens, 4-Sep-2018.)

Assertion

Ref	Expression
csbprg	⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵})

Proof of Theorem csbprg

Step	Hyp	Ref	Expression
1		csbun 4389	. . 3 ⊢ ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) = (⦋𝐶 / 𝑥⦌{𝐴} ∪ ⦋𝐶 / 𝑥⦌{𝐵})
2		csbsng 4661	. . . 4 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴} = {⦋𝐶 / 𝑥⦌𝐴})
3		csbsng 4661	. . . 4 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐵} = {⦋𝐶 / 𝑥⦌𝐵})
4	2, 3	uneq12d 4117	. . 3 ⊢ (𝐶 ∈ 𝑉 → (⦋𝐶 / 𝑥⦌{𝐴} ∪ ⦋𝐶 / 𝑥⦌{𝐵}) = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵}))
5	1, 4	eqtrid 2803	. 2 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵}))
6		df-pr 4579	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
7	6	csbeq2i 3855	. 2 ⊢ ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵})
8		df-pr 4579	. 2 ⊢ {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵} = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵})
9	5, 7, 8	3eqtr4g 2816	1 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵})

Syntax hints:   → wi 4   = wceq 1554   ∈ wcel 2136  ⦋csb 3847   ∪ cun 3897  {csn 4576  {cpr 4578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-nul 4281  df-sn 4577  df-pr 4579

This theorem is referenced by:  csbopg  4843