Theorem csbexa 3886

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 3885	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
3	1, 2	mpan 400	. 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
4		csbexa.2	. 2 ⊢ 𝐵 ∈ V
5	3, 4	mpg 1340	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V

Syntax hints:  ∀wal 1241   ∈ wcel 1393  Vcvv 2557  ⦋csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by:  dfmpt2  5844