Theorem difexd 4272

Description: Existence of a difference. (Contributed by SN, 16-Jul-2024.)

Hypothesis

Ref	Expression
difexd.1	|- ( ph -> A e. V )

Assertion

Ref	Expression
difexd	|- ( ph -> ( A \ B ) e. _V )

Proof of Theorem difexd

Step	Hyp	Ref	Expression
1		difexd.1	. 2 |- ( ph -> A e. V )
2		difexg 4270	. 2 |- ( A e. V -> ( A \ B ) e. _V )
3	1, 2	syl 14	1 |- ( ph -> ( A \ B ) e. _V )

Syntax hints:   -> wi 4   e. wcel 2209   _Vcvv 2821   \ cdif 3217

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-in1 623  ax-in2 624  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  ax-sep 4244

This theorem 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-dif 3222  df-in 3226  df-ss 3233

This theorem is referenced by:  hashf1lem1  11263