Theorem rabbidv 1809

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

Hypothesis

Ref	Expression
rabbidv.1	|- ((ph /\ x e. A) -> (ps <-> ch))

Assertion

Ref	Expression
rabbidv	|- (ph -> {x e. A | ps} = {x e. A | ch})

Distinct variable group:   ph,x

Proof of Theorem rabbidv

Step	Hyp	Ref	Expression
1		rabbidv.1	. . . 4 |- ((ph /\ x e. A) -> (ps <-> ch))
2	1	pm5.32da 651	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. A /\ ch)))
3	2	abbidv 1580	. 2 |- (ph -> {x | (x e. A /\ ps)} = {x | (x e. A /\ ch)})
4		df-rab 1655	. 2 |- {x e. A | ps} = {x | (x e. A /\ ps)}
5		df-rab 1655	. 2 |- {x e. A | ch} = {x | (x e. A /\ ch)}
6	3, 4, 5	3eqtr4g 1534	1 |- (ph -> {x e. A | ps} = {x e. A | ch})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  {crab 1651

This theorem is referenced by:  rabbisdv 1810  onsucmin 3078  dfinfmr 6069  infmsup 6070  supxrre 6085  iooint 6373  cncnplem4 7774  blin 7849  addinv 8124  ee7.2a 10420

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rab 1655

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