Theorem csbex 3147

Description: The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

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	. . 3 ⊢ A ∈ V
2		csbexg 3146	. . 3 ⊢ ((A ∈ V ∧ ∀x B ∈ V) → [A / x]B ∈ V)
3	1, 2	mpan 651	. 2 ⊢ (∀x B ∈ V → [A / x]B ∈ V)
4		csbex.2	. 2 ⊢ B ∈ V
5	3, 4	mpg 1548	1 ⊢ [A / x]B ∈ V

Syntax hints:  ∀wal 1540   ∈ wcel 1710  Vcvv 2859  [csb 3136

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 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by: (None)