Theorem bnj13 12378

Description: First-order logic and set theory.

Assertion

Ref	Expression
bnj13	|- (y e. {x e. A | ph} <-> (y e. A /\ [y / x]ph))

Distinct variable group:   x,A

Proof of Theorem bnj13

Step	Hyp	Ref	Expression
1		df-rab 2112	. . . 4 |- {x e. A | ph} = {x | (x e. A /\ ph)}
2	1	eleq2i 1961	. . 3 |- (y e. {x e. A | ph} <-> y e. {x | (x e. A /\ ph)})
3		df-clab 1872	. . 3 |- (y e. {x | (x e. A /\ ph)} <-> [y / x](x e. A /\ ph))
4		sban 1607	. . 3 |- ([y / x](x e. A /\ ph) <-> ([y / x]x e. A /\ [y / x]ph))
5	2, 3, 4	3bitri 194	. 2 |- (y e. {x e. A | ph} <-> ([y / x]x e. A /\ [y / x]ph))
6		clelsb3 1990	. . 3 |- ([y / x]x e. A <-> y e. A)
7	6	anbi1i 539	. 2 |- (([y / x]x e. A /\ [y / x]ph) <-> (y e. A /\ [y / x]ph))
8	5, 7	bitri 190	1 |- (y e. {x e. A | ph} <-> (y e. A /\ [y / x]ph))

Syntax hints:   <-> wb 163   /\ wa 240   e. wcel 1300  [wsbc 1534  {cab 1871  {crab 2108

This theorem is referenced by:  bnj15 12379  bnj54 12428

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-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-rab 2112

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