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Theorem r19.36av 2759
 Description: One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. The other direction doesn't hold when A is empty. (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
r19.36av (x A (φψ) → (x A φψ))
Distinct variable group:   ψ,x
Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem r19.36av
StepHypRef Expression
1 r19.35 2758 . 2 (x A (φψ) ↔ (x A φx A ψ))
2 idd 21 . . . 4 (x A → (ψψ))
32rexlimiv 2732 . . 3 (x A ψψ)
43imim2i 13 . 2 ((x A φx A ψ) → (x A φψ))
51, 4sylbi 187 1 (x A (φψ) → (x A φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2619  df-rex 2620 This theorem is referenced by:  iinss  4017
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