Theorem bj-elssuniab 11033

Description: Version of elssuni 3655 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 2833	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )
2		elssuni 3655	. 2 |- ( A e. { x | ph } -> A C_ U. { x | ph } )
3	1, 2	syl6bi 161	1 |- ( A e. V -> ( [. A / x ]. ph -> A C_ U. { x | ph } ) )

Syntax hints:   -> wi 4   e. wcel 1434   {cab 2069   F/_wnfc 2210   [.wsbc 2826   C_ wss 2984   U.cuni 3627

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-sbc 2827  df-in 2990  df-ss 2997  df-uni 3628

This theorem is referenced by: (None)