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Theorem r19.45av 2768
 Description: Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.45av (x A (φ ψ) → (φ x A ψ))
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   A(x)

Proof of Theorem r19.45av
StepHypRef Expression
1 r19.43 2766 . 2 (x A (φ ψ) ↔ (x A φ x A ψ))
2 idd 21 . . . 4 (x A → (φφ))
32rexlimiv 2732 . . 3 (x A φφ)
43orim1i 503 . 2 ((x A φ x A ψ) → (φ x A ψ))
51, 4sylbi 187 1 (x A (φ ψ) → (φ x A ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357   ∈ wcel 1710  ∃wrex 2615 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-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2619  df-rex 2620 This theorem is referenced by: (None)
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