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Theorem ralbiim 2751
 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 609 . . 3 ((φψ) ↔ ((φψ) (ψφ)))
21ralbii 2638 . 2 (x A (φψ) ↔ x A ((φψ) (ψφ)))
3 r19.26 2746 . 2 (x A ((φψ) (ψφ)) ↔ (x A (φψ) x A (ψφ)))
42, 3bitri 240 1 (x A (φψ) ↔ (x A (φψ) x A (ψφ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2619 This theorem is referenced by:  eqreu  3028  ssofeq  4077
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