Theorem csbexg 2004

Description: The existence of proper substitution into a class.

Assertion

Ref	Expression
csbexg	⊢ ((A ∈ C ⋀ ∀x B ∈ D) → [A / x]B ∈ V)

Proof of Theorem csbexg

Step	Hyp	Ref	Expression
1		a4sbc 1941	. . . . 5 ⊢ (A ∈ C → (∀x{y∣y ∈ B} ∈ V → [A / x]{y∣y ∈ B} ∈ V))
2		elisset 1813	. . . . . . 7 ⊢ (B ∈ D → B ∈ V)
3		abid2 1577	. . . . . . 7 ⊢ {y∣y ∈ B} = B
4	2, 3	syl5eqel 1549	. . . . . 6 ⊢ (B ∈ D → {y∣y ∈ B} ∈ V)
5	4	19.20i 990	. . . . 5 ⊢ (∀x B ∈ D → ∀x{y∣y ∈ B} ∈ V)
6	1, 5	syl5 21	. . . 4 ⊢ (A ∈ C → (∀x B ∈ D → [A / x]{y∣y ∈ B} ∈ V))
7	6	imp 350	. . 3 ⊢ ((A ∈ C ⋀ ∀x B ∈ D) → [A / x]{y∣y ∈ B} ∈ V)
8		ax-17 969	. . . . 5 ⊢ (y ∈ V → ∀x y ∈ V)
9	8	sbcabel 1992	. . . 4 ⊢ (A ∈ C → ([A / x]{y∣y ∈ B} ∈ V ↔ {y∣[A / x]y ∈ B} ∈ V))
10	9	adantr 389	. . 3 ⊢ ((A ∈ C ⋀ ∀x B ∈ D) → ([A / x]{y∣y ∈ B} ∈ V ↔ {y∣[A / x]y ∈ B} ∈ V))
11	7, 10	mpbid 195	. 2 ⊢ ((A ∈ C ⋀ ∀x B ∈ D) → {y∣[A / x]y ∈ B} ∈ V)
12		df-csb 1998	. 2 ⊢ [A / x]B = {y∣[A / x]y ∈ B}
13	11, 12	syl5eqel 1549	1 ⊢ ((A ∈ C ⋀ ∀x B ∈ D) → [A / x]B ∈ V)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 952   ∈ wcel 956  [wsbc 1168  {cab 1461  Vcvv 1807  [csb 1997

This theorem is referenced by:  csbex 2005  csbnestglem 2031  csbnestg 2032  csbnest1g 2033  sbcnestg 2034

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938  df-csb 1998

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