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Theorem ralim 2249
 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 2184 . . 3 (x A (φψ) ↔ x(x A → (φψ)))
2 ax-2 6 . . . 4 ((x A → (φψ)) → ((x Aφ) → (x Aψ)))
32al2imi 1284 . . 3 (x(x A → (φψ)) → (x(x Aφ) → x(x Aψ)))
41, 3sylbi 112 . 2 (x A (φψ) → (x(x Aφ) → x(x Aψ)))
5 df-ral 2184 . 2 (x A φx(x Aφ))
6 df-ral 2184 . 2 (x A ψx(x Aψ))
74, 5, 63imtr4g 192 1 (x A (φψ) → (x A φx A ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1272   ∈ wcel 1332  ∀wral 2179 This theorem is referenced by:  ral2imi  2254 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-5 1273  ax-gen 1275 This theorem depends on definitions:  df-bi 108  df-ral 2184
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