Theorem csbexg 3147

Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbexg	⊢ ((A ∈ V ∧ ∀x B ∈ W) → [A / x]B ∈ V)

Proof of Theorem csbexg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3138	. 2 ⊢ [A / x]B = {y ∣ [̣A / x]̣y ∈ B}
2		abid2 2471	. . . . . . 7 ⊢ {y ∣ y ∈ B} = B
3		elex 2868	. . . . . . 7 ⊢ (B ∈ W → B ∈ V)
4	2, 3	syl5eqel 2437	. . . . . 6 ⊢ (B ∈ W → {y ∣ y ∈ B} ∈ V)
5	4	alimi 1559	. . . . 5 ⊢ (∀x B ∈ W → ∀x{y ∣ y ∈ B} ∈ V)
6		spsbc 3059	. . . . 5 ⊢ (A ∈ V → (∀x{y ∣ y ∈ B} ∈ V → [̣A / x]̣{y ∣ y ∈ B} ∈ V))
7	5, 6	syl5 28	. . . 4 ⊢ (A ∈ V → (∀x B ∈ W → [̣A / x]̣{y ∣ y ∈ B} ∈ V))
8	7	imp 418	. . 3 ⊢ ((A ∈ V ∧ ∀x B ∈ W) → [̣A / x]̣{y ∣ y ∈ B} ∈ V)
9		nfcv 2490	. . . . 5 ⊢ ℲxV
10	9	sbcabel 3124	. . . 4 ⊢ (A ∈ V → ([̣A / x]̣{y ∣ y ∈ B} ∈ V ↔ {y ∣ [̣A / x]̣y ∈ B} ∈ V))
11	10	adantr 451	. . 3 ⊢ ((A ∈ V ∧ ∀x B ∈ W) → ([̣A / x]̣{y ∣ y ∈ B} ∈ V ↔ {y ∣ [̣A / x]̣y ∈ B} ∈ V))
12	8, 11	mpbid 201	. 2 ⊢ ((A ∈ V ∧ ∀x B ∈ W) → {y ∣ [̣A / x]̣y ∈ B} ∈ V)
13	1, 12	syl5eqel 2437	1 ⊢ ((A ∈ V ∧ ∀x B ∈ W) → [A / x]B ∈ V)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  Vcvv 2860  [̣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:  csbex  3148