Theorem bnj1454 13135

Description: First-order logic and set theory.

Hypothesis

Ref	Expression
bnj1454.1	|- A = {x | ph}

Assertion

Ref	Expression
bnj1454	|- (B e. _V -> (B e. A <-> [B / x]ph))

Proof of Theorem bnj1454

Step	Hyp	Ref	Expression
1		elabsg 2488	. 2 |- (B e. _V -> (B e. {x | ph} <-> [B / x]ph))
2		bnj1454.1	. . 3 |- A = {x | ph}
3	2	eleq2i 1961	. 2 |- (B e. A <-> B e. {x | ph})
4	1, 3	syl5bb 591	1 |- (B e. _V -> (B e. A <-> [B / x]ph))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  [wsbc 1534  {cab 1871  _Vcvv 2292

This theorem is referenced by:  bnj1462 13546  bnj1463 13550

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-rab 2112  df-v 2294  df-sbc 2454

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