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Theorem ralim 2685
Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
Assertion
Ref Expression
ralim (x A (φψ) → (x A φx A ψ))

Proof of Theorem ralim
StepHypRef Expression
1 df-ral 2619 . . 3 (x A (φψ) ↔ x(x A → (φψ)))
2 ax-2 7 . . . 4 ((x A → (φψ)) → ((x Aφ) → (x Aψ)))
32al2imi 1561 . . 3 (x(x A → (φψ)) → (x(x Aφ) → x(x Aψ)))
41, 3sylbi 187 . 2 (x A (φψ) → (x(x Aφ) → x(x Aψ)))
5 df-ral 2619 . 2 (x A φx(x Aφ))
6 df-ral 2619 . 2 (x A ψx(x Aψ))
74, 5, 63imtr4g 261 1 (x A (φψ) → (x A φx A ψ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540   wcel 1710  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
This theorem depends on definitions:  df-bi 177  df-ral 2619
This theorem is referenced by:  ral2imi  2690  r19.30  2756
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