Theorem csbexa 4052

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 4051	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
3	1, 2	mpan 420	. 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
4		csbexa.2	. 2 ⊢ 𝐵 ∈ V
5	3, 4	mpg 1427	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V

Syntax hints:  ∀wal 1329   ∈ wcel 1480  Vcvv 2681  ⦋csb 2998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999

This theorem is referenced by:  dfmpo  6113