Theorem csbsng 4652

Description: Distribute proper substitution through the singleton of a class. csbsng 4652 is derived from the virtual deduction proof csbsngVD 45319. (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 4380	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵}
2		sbceq2g 4359	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵))
3	2	abbidv 2802	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵})
4	1, 3	eqtrid 2783	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵})
5		df-sn 4568	. . 3 ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
6	5	csbeq2i 3845	. 2 ⊢ ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}
7		df-sn 4568	. 2 ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}
8	4, 6, 7	3eqtr4g 2796	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {cab 2714  [wsbc 3728  ⦋csb 3837  {csn 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-nul 4274  df-sn 4568

This theorem is referenced by:  csbprg  4653  csbopg  4834  csbpredg  6271  csbfv12gALTVD  45325