Theorem sbc8gOLD 2478

Description: This is the closest we can get to df-sbc 2454 if we start from dfsbcq 2455 (see its comments).

Assertion

Ref	Expression
sbc8gOLD	|- (A e. B -> ([A / x]ph <-> A e. {x | ph}))

Proof of Theorem sbc8gOLD

Step	Hyp	Ref	Expression
1		sbc7g 2476	. 2 |- (A e. B -> ([A / x]ph <-> E.y(y = A /\ [y / x]ph)))
2		df-clab 1872	. . . . . 6 |- (y e. {x | ph} <-> [y / x]ph)
3	2	anbi2i 538	. . . . 5 |- ((y = A /\ y e. {x | ph}) <-> (y = A /\ [y / x]ph))
4	3	exbii 1398	. . . 4 |- (E.y(y = A /\ y e. {x | ph}) <-> E.y(y = A /\ [y / x]ph))
5	4	a1i 8	. . 3 |- (A e. B -> (E.y(y = A /\ y e. {x | ph}) <-> E.y(y = A /\ [y / x]ph)))
6		df-clel 1880	. . 3 |- (A e. {x | ph} <-> E.y(y = A /\ y e. {x | ph}))
7	5, 6	syl5rbb 592	. 2 |- (A e. B -> (E.y(y = A /\ [y / x]ph) <-> A e. {x | ph}))
8	1, 7	bitrd 587	1 |- (A e. B -> ([A / x]ph <-> A e. {x | ph}))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  {cab 1871

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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