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Theorem r19.32r 2451
 Description: One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence. (Contributed by Jim Kingdon, 19-Aug-2018.)
Hypothesis
Ref Expression
r19.32r.1 xφ
Assertion
Ref Expression
r19.32r ((φ x A ψ) → x A (φ ψ))

Proof of Theorem r19.32r
StepHypRef Expression
1 r19.32r.1 . . . 4 xφ
2 orc 632 . . . . 5 (φ → (φ ψ))
32a1d 22 . . . 4 (φ → (x A → (φ ψ)))
41, 3alrimi 1412 . . 3 (φx(x A → (φ ψ)))
5 df-ral 2305 . . . 4 (x A ψx(x Aψ))
6 olc 631 . . . . . 6 (ψ → (φ ψ))
76imim2i 12 . . . . 5 ((x Aψ) → (x A → (φ ψ)))
87alimi 1341 . . . 4 (x(x Aψ) → x(x A → (φ ψ)))
95, 8sylbi 114 . . 3 (x A ψx(x A → (φ ψ)))
104, 9jaoi 635 . 2 ((φ x A ψ) → x(x A → (φ ψ)))
11 df-ral 2305 . 2 (x A (φ ψ) ↔ x(x A → (φ ψ)))
1210, 11sylibr 137 1 ((φ x A ψ) → x A (φ ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 628  ∀wal 1240  Ⅎwnf 1346   ∈ 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-io 629  ax-5 1333  ax-gen 1335  ax-4 1397 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305 This theorem is referenced by:  r19.32vr  2452
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