Theorem bj-elssuniab 16155

Description: Version of elssuni 3916 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

Hypothesis

Ref	Expression
bj-elssuniab.nf	|- F/_ x A

Assertion

Ref	Expression
bj-elssuniab	|- ( A e. V -> ( [. A / x ]. ph -> A C_ U. { x | ph } ) )

Proof of Theorem bj-elssuniab

Step	Hyp	Ref	Expression
1		sbc8g 3036	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )
2		elssuni 3916	. 2 |- ( A e. { x | ph } -> A C_ U. { x | ph } )
3	1, 2	biimtrdi 163	1 |- ( A e. V -> ( [. A / x ]. ph -> A C_ U. { x | ph } ) )

Syntax hints:   -> wi 4   e. wcel 2200   {cab 2215   F/_wnfc 2359   [.wsbc 3028   C_ wss 3197   U.cuni 3888

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-in 3203  df-ss 3210  df-uni 3889

This theorem is referenced by: (None)