Theorem csbprg 4670

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 4402	. . 3 ⊢ ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) = (⦋𝐶 / 𝑥⦌{𝐴} ∪ ⦋𝐶 / 𝑥⦌{𝐵})
2		csbsng 4669	. . . 4 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴} = {⦋𝐶 / 𝑥⦌𝐴})
3		csbsng 4669	. . . 4 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐵} = {⦋𝐶 / 𝑥⦌𝐵})
4	2, 3	uneq12d 4171	. . 3 ⊢ (𝐶 ∈ 𝑉 → (⦋𝐶 / 𝑥⦌{𝐴} ∪ ⦋𝐶 / 𝑥⦌{𝐵}) = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵}))
5	1, 4	eqtrid 2810	. 2 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵}))
6		df-pr 4591	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
7	6	csbeq2i 3859	. 2 ⊢ ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵})
8		df-pr 4591	. 2 ⊢ {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵} = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵})
9	5, 7, 8	3eqtr4g 2823	1 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ⦋csb 3851   ∪ cun 4152  {csn 4588  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-v 3456  df-sbc 3743  df-csb 3852  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591

This theorem is used by:  csbopg  4854