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Theorem r19.27av 2752
 Description: Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
r19.27av ((x A φ ψ) → x A (φ ψ))
Distinct variable group:   ψ,x
Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem r19.27av
StepHypRef Expression
1 ax-1 6 . . . 4 (ψ → (x Aψ))
21ralrimiv 2696 . . 3 (ψx A ψ)
32anim2i 552 . 2 ((x A φ ψ) → (x A φ x A ψ))
4 r19.26 2746 . 2 (x A (φ ψ) ↔ (x A φ x A ψ))
53, 4sylibr 203 1 ((x A φ ψ) → x A (φ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ 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  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619 This theorem is referenced by:  r19.28av  2753
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