Theorem csbsng 4688

Description: Distribute proper substitution through the singleton of a class. csbsng 4688 is derived from the virtual deduction proof csbsngVD 44846. (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 4420	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵}
2		sbceq2g 4399	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵))
3	2	abbidv 2800	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵})
4	1, 3	eqtrid 2781	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵})
5		df-sn 4607	. . 3 ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
6	5	csbeq2i 3887	. 2 ⊢ ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}
7		df-sn 4607	. 2 ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}
8	4, 6, 7	3eqtr4g 2794	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  {cab 2712  [wsbc 3770  ⦋csb 3879  {csn 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-nul 4314  df-sn 4607

This theorem is referenced by:  csbprg  4689  csbopg  4871  csbpredg  6307  csbfv12gALTVD  44852