Theorem csbex 1999

Description: The existence of proper substitution into a class.

Hypotheses

Ref	Expression
csbex.1	⊢ A ∈ V
csbex.2	⊢ B ∈ V

Assertion

Ref	Expression
csbex	⊢ [A / x]B ∈ V

Proof of Theorem csbex

Step	Hyp	Ref	Expression
1		csbex.1	. 2 ⊢ A ∈ V
2		csbex.2	. . 3 ⊢ B ∈ V
3	2	ax-gen 960	. 2 ⊢ ∀x B ∈ V
4		csbexg 1998	. 2 ⊢ ((A ∈ V ⋀ ∀x B ∈ V) → [A / x]B ∈ V)
5	1, 3, 4	mp2an 695	1 ⊢ [A / x]B ∈ V

Syntax hints:  ∀wal 951   ∈ wcel 955  Vcvv 1802  [csb 1991

This theorem is referenced by:  fvopab4sf 3767  fvopabs 3777  fopabcos 3818  fsum1slem 6946  fsump1f 6949  fsump1slem 6950  csbfsumlem 6964

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932  df-csb 1992

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