Theorem csbsng 4672

Description: Distribute proper substitution through the singleton of a class. csbsng 4672 is derived from the virtual deduction proof csbsngVD 45702. (Contributed by Alan Sare, 10-Nov-2012.)

Assertion

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

Proof of Theorem csbsng

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbab 4401	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵}
2		sbceq2g 4380	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵))
3	2	abbidv 2828	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵})
4	1, 3	eqtrid 2809	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵})
5		df-sn 4588	. . 3 ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
6	5	csbeq2i 3858	. 2 ⊢ ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}
7		df-sn 4588	. 2 ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}
8	4, 6, 7	3eqtr4g 2822	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {cab 2740  [wsbc 3742  ⦋csb 3850  {csn 4587

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-nul 4283  df-sn 4588

This theorem is used by:  csbprg  4673  csbopg  4854  csbpredg  6309  csbfv12gALTVD  45708