Theorem rabbii 1796

Description: Equivalent wff's yield equal restricted class abstractions (inference rule).

Hypothesis

Ref	Expression
rabbii.1	|- (x e. A -> (ps <-> ch))

Assertion

Ref	Expression
rabbii	|- {x e. A | ps} = {x e. A | ch}

Proof of Theorem rabbii

Step	Hyp	Ref	Expression
1		rabbii.1	. . . 4 |- (x e. A -> (ps <-> ch))
2	1	pm5.32i 643	. . 3 |- ((x e. A /\ ps) <-> (x e. A /\ ch))
3	2	abbii 1567	. 2 |- {x | (x e. A /\ ps)} = {x | (x e. A /\ ch)}
4		df-rab 1644	. 2 |- {x e. A | ps} = {x | (x e. A /\ ps)}
5		df-rab 1644	. 2 |- {x e. A | ch} = {x | (x e. A /\ ch)}
6	3, 4, 5	3eqtr4 1497	1 |- {x e. A | ps} = {x e. A | ch}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  {crab 1640

This theorem is referenced by:  rabxfr 2892  reuunixfr 2896  bm2.5ii 3009  nlimon 3112  rankval2 4642  ranksn 4661  hta 4700  kmlem3 4739  infmsup 6015  dfuz 6150  ioopos 6326  isupivth 7225  dsupivthlem 7226  ivth2OLD 7234  alephsuc3 7527  spwval2 8577  spwval3 8578  spwnex3 8579  pilem3 8592  eff1o 8670  dfbdop2 9703  hhblo 9745  cnlnadjlem5 9919  cdj3lem3 10270  cdj3lem3b 10272

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-rab 1644

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