Theorem csbabg 3198

Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
csbabg	⊢ (A ∈ V → [A / x]{y ∣ φ} = {y ∣ [̣A / x]̣φ})

Distinct variable groups:   y,A   x,y

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

Proof of Theorem csbabg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbccom 3118	. . . 4 ⊢ ([̣z / y]̣[̣A / x]̣φ ↔ [̣A / x]̣[̣z / y]̣φ)
2		df-clab 2340	. . . . 5 ⊢ (z ∈ {y ∣ [̣A / x]̣φ} ↔ [z / y][̣A / x]̣φ)
3		sbsbc 3051	. . . . 5 ⊢ ([z / y][̣A / x]̣φ ↔ [̣z / y]̣[̣A / x]̣φ)
4	2, 3	bitri 240	. . . 4 ⊢ (z ∈ {y ∣ [̣A / x]̣φ} ↔ [̣z / y]̣[̣A / x]̣φ)
5		df-clab 2340	. . . . . 6 ⊢ (z ∈ {y ∣ φ} ↔ [z / y]φ)
6		sbsbc 3051	. . . . . 6 ⊢ ([z / y]φ ↔ [̣z / y]̣φ)
7	5, 6	bitri 240	. . . . 5 ⊢ (z ∈ {y ∣ φ} ↔ [̣z / y]̣φ)
8	7	sbcbii 3102	. . . 4 ⊢ ([̣A / x]̣z ∈ {y ∣ φ} ↔ [̣A / x]̣[̣z / y]̣φ)
9	1, 4, 8	3bitr4i 268	. . 3 ⊢ (z ∈ {y ∣ [̣A / x]̣φ} ↔ [̣A / x]̣z ∈ {y ∣ φ})
10		sbcel2g 3158	. . 3 ⊢ (A ∈ V → ([̣A / x]̣z ∈ {y ∣ φ} ↔ z ∈ [A / x]{y ∣ φ}))
11	9, 10	syl5rbb 249	. 2 ⊢ (A ∈ V → (z ∈ [A / x]{y ∣ φ} ↔ z ∈ {y ∣ [̣A / x]̣φ}))
12	11	eqrdv 2351	1 ⊢ (A ∈ V → [A / x]{y ∣ φ} = {y ∣ [̣A / x]̣φ})

Syntax hints:   → wi 4   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  [̣wsbc 3047  [csb 3137

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

This theorem is referenced by:  csbsng  3786  csbunig  3900  csbxpg  4814  csbrng  4967