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Theorem ralbiim 2441
Description: Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)
Assertion
Ref Expression
ralbiim (x A (φψ) ↔ (x A (φψ) x A (ψφ)))

Proof of Theorem ralbiim
StepHypRef Expression
1 dfbi2 368 . . 3 ((φψ) ↔ ((φψ) (ψφ)))
21ralbii 2324 . 2 (x A (φψ) ↔ x A ((φψ) (ψφ)))
3 r19.26 2435 . 2 (x A ((φψ) (ψφ)) ↔ (x A (φψ) x A (ψφ)))
42, 3bitri 173 1 (x A (φψ) ↔ (x A (φψ) x A (ψφ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  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  ax-4 1397  ax-17 1416
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-ral 2305
This theorem is referenced by:  eqreu  2727
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