Theorem bj-elssuniab 12998

Description: Version of elssuni 3764 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 2916	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )
2		elssuni 3764	. 2 |- ( A e. { x | ph } -> A C_ U. { x | ph } )
3	1, 2	syl6bi 162	1 |- ( A e. V -> ( [. A / x ]. ph -> A C_ U. { x | ph } ) )

Syntax hints:   -> wi 4   e. wcel 1480   {cab 2125   F/_wnfc 2268   [.wsbc 2909   C_ wss 3071   U.cuni 3736

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 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-in 3077  df-ss 3084  df-uni 3737

This theorem is referenced by: (None)