Theorem csbexa 4262

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 4261	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
3	1, 2	mpan 428	. 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V)
4		csbexa.2	. 2 ⊢ 𝐵 ∈ V
5	3, 4	mpg 1504	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V

This proof depends on syntax axioms:  ∀wal 1400   ∈ wcel 2209  Vcvv 2821  ⦋csb 3147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-csb 3148

This theorem is used by:  dfmpo  6459  rhmex  14466  fnpsr  15053  fnmpl  15086