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Theorem ralbii2 2328
Description: Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
Hypothesis
Ref Expression
ralbii2.1 ((x Aφ) ↔ (x Bψ))
Assertion
Ref Expression
ralbii2 (x A φx B ψ)

Proof of Theorem ralbii2
StepHypRef Expression
1 ralbii2.1 . . 3 ((x Aφ) ↔ (x Bψ))
21albii 1356 . 2 (x(x Aφ) ↔ x(x Bψ))
3 df-ral 2305 . 2 (x A φx(x Aφ))
4 df-ral 2305 . 2 (x B ψx(x Bψ))
52, 3, 43bitr4i 201 1 (x A φx B ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   wcel 1390  wral 2300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335
This theorem depends on definitions:  df-bi 110  df-ral 2305
This theorem is referenced by:  raleqbii  2330  ralbiia  2332  ralrab  2696  raldifb  3077  raluz2  8298  ralrp  8379
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