Theorem csbsng 4434

Description: Distribute proper substitution through the singleton of a class. csbsng 4434 is derived from the virtual deduction proof csbsngVD 39884. (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 4205	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵}
2		sbceq2g 4186	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵))
3	2	abbidv 2919	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵})
4	1, 3	syl5eq 2846	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵})
5		df-sn 4370	. . 3 ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵}
6	5	csbeq2i 4189	. 2 ⊢ ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}
7		df-sn 4370	. 2 ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}
8	4, 6, 7	3eqtr4g 2859	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})

Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157  {cab 2786  [wsbc 3634  ⦋csb 3729  {csn 4369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-nul 4117  df-sn 4370

This theorem is referenced by:  csbprg  4435  csbopg  4612  csbpredg  33670  csbfv12gALTVD  39890