Theorem rabbidv 1806

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 649	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. A /\ ch)))
3	2	abbidv 1577	. 2 |- (ph -> {x | (x e. A /\ ps)} = {x | (x e. A /\ ch)})
4		df-rab 1652	. 2 |- {x e. A | ps} = {x | (x e. A /\ ps)}
5		df-rab 1652	. 2 |- {x e. A | ch} = {x | (x e. A /\ ch)}
6	3, 4, 5	3eqtr4g 1531	1 |- (ph -> {x e. A | ps} = {x e. A | ch})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  {crab 1648

This theorem is referenced by:  rabbisdv 1807  onsucmin 3072  dfinfmr 6067  infmsup 6068  supxrre 6083  iooint 6372  cncnplem4 7777  blin 7852  addinv 8128  ee7.2a 10425

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-10 966  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-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652

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