Theorem bj-elssuniab 15731

Description: Version of elssuni 3878 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 3006	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )
2		elssuni 3878	. 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 2176   {cab 2191   F/_wnfc 2335   [.wsbc 2998   C_ wss 3166   U.cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999  df-in 3172  df-ss 3179  df-uni 3851

This theorem is referenced by: (None)