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Theorem r19.27av 2442
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 5 . . . 4 (ψ → (x Aψ))
21ralrimiv 2385 . . 3 (ψx A ψ)
32anim2i 324 . 2 ((x A φ ψ) → (x A φ x A ψ))
4 r19.26 2435 . 2 (x A (φ ψ) ↔ (x A φ x A ψ))
53, 4sylibr 137 1 ((x A φ ψ) → x A (φ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  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-nf 1347  df-ral 2305
This theorem is referenced by:  r19.28av  2443
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