Theorem csbsng 3786

Description: Distribute proper substitution through the singleton of a class. csbsng 3786 is derived from the virtual deduction proof csbsngVD in set.mm. (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		csbabg 3198	. . 3 ⊢ (A ∈ V → [A / x]{y ∣ y = B} = {y ∣ [̣A / x]̣y = B})
2		sbceq2g 3159	. . . 4 ⊢ (A ∈ V → ([̣A / x]̣y = B ↔ y = [A / x]B))
3	2	abbidv 2468	. . 3 ⊢ (A ∈ V → {y ∣ [̣A / x]̣y = B} = {y ∣ y = [A / x]B})
4	1, 3	eqtrd 2385	. 2 ⊢ (A ∈ V → [A / x]{y ∣ y = B} = {y ∣ y = [A / x]B})
5		df-sn 3742	. . 3 ⊢ {B} = {y ∣ y = B}
6	5	csbeq2i 3163	. 2 ⊢ [A / x]{B} = [A / x]{y ∣ y = B}
7		df-sn 3742	. 2 ⊢ {[A / x]B} = {y ∣ y = [A / x]B}
8	4, 6, 7	3eqtr4g 2410	1 ⊢ (A ∈ V → [A / x]{B} = {[A / x]B})

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3047  [csb 3137  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138  df-sn 3742

This theorem is referenced by: (None)