Theorem bj-elssuniab 16509

Description: Version of elssuni 3926 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 3040	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A e. { x | ph } ) )
2		elssuni 3926	. 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 2202   {cab 2217   F/_wnfc 2362   [.wsbc 3032   C_ wss 3201   U.cuni 3898

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sbc 3033  df-in 3207  df-ss 3214  df-uni 3899

This theorem is referenced by: (None)