Theorem csbexa 4177

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
csbexa.1	⊢ 𝐴 ∈ V
csbexa.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
csbexa	⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V

Proof of Theorem csbexa

Step	Hyp	Ref	Expression
1		csbexa.1	. . 3 ⊢ 𝐴 ∈ V
2		csbexga 4176	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
3	1, 2	mpan 424	. 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
4		csbexa.2	. 2 ⊢ 𝐵 ∈ V
5	3, 4	mpg 1475	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V

Syntax hints:  ∀wal 1371   ∈ wcel 2177  Vcvv 2773  ⦋csb 3094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3000  df-csb 3095

This theorem is referenced by:  dfmpo  6316  rhmex  13963  fnpsr  14473  fnmpl  14499