Theorem elabsg 1965

Description: Membership in a class abstraction, expressed in terms of class substitution. Conveniently, this theorem has no distinct variable restrictions. Except for the antecedent, this theorem is "almost" like df-sbc 1942 but was proved using only dfsbcq 1943 as its starting point (making no other reference to df-sbc 1942). We prefer not to make direct reference to df-sbc 1942 (i.e. commit to it) since its behavior at proper classes is at odds with Quine, whereas dfsbcq 1943 is not. (Quine's class substitution cannot be expressed in closed form.) This theorem serves as a weaker Quine-compatible substitute for df-sbc 1942.

Assertion

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

Proof of Theorem elabsg

Step	Hyp	Ref	Expression
1		elisset 1817	. 2 |- (A e. B -> A e. V)
2		elabs2 1964	. . 3 |- (A e. {x | ph} <-> (A e. V /\ [A / x]ph))
3	2	baib 685	. 2 |- (A e. V -> (A e. {x | ph} <-> [A / x]ph))
4	1, 3	syl 10	1 |- (A e. B -> (A e. {x | ph} <-> [A / x]ph))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 958  [wsbc 1170  {cab 1463  Vcvv 1811

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-v 1812  df-sbc 1942

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