Theorem csbopabg 4638

Description: Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
csbopabg	⊢ (A ∈ V → [A / x]{⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [̣A / x]̣φ})

Distinct variable groups:   y,z,A   x,y,z

Allowed substitution hints:   φ(x,y,z)   A(x)   V(x,y,z)

Proof of Theorem csbopabg

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3140	. . 3 ⊢ (w = A → [w / x]{⟨y, z⟩ ∣ φ} = [A / x]{⟨y, z⟩ ∣ φ})
2		dfsbcq2 3050	. . . 4 ⊢ (w = A → ([w / x]φ ↔ [̣A / x]̣φ))
3	2	opabbidv 4626	. . 3 ⊢ (w = A → {⟨y, z⟩ ∣ [w / x]φ} = {⟨y, z⟩ ∣ [̣A / x]̣φ})
4	1, 3	eqeq12d 2367	. 2 ⊢ (w = A → ([w / x]{⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [w / x]φ} ↔ [A / x]{⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [̣A / x]̣φ}))
5		vex 2863	. . 3 ⊢ w ∈ V
6		nfs1v 2106	. . . 4 ⊢ Ⅎx[w / x]φ
7	6	nfopab 4628	. . 3 ⊢ Ⅎx{⟨y, z⟩ ∣ [w / x]φ}
8		sbequ12 1919	. . . 4 ⊢ (x = w → (φ ↔ [w / x]φ))
9	8	opabbidv 4626	. . 3 ⊢ (x = w → {⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [w / x]φ})
10	5, 7, 9	csbief 3178	. 2 ⊢ [w / x]{⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [w / x]φ}
11	4, 10	vtoclg 2915	1 ⊢ (A ∈ V → [A / x]{⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [̣A / x]̣φ})

Syntax hints:   → wi 4   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3047  [csb 3137  {copab 4623

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-3an 936  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-opab 4624

This theorem is referenced by: (None)