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Theorem r19.32r 2435
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 620 . . . . 5 (φ → (φ ψ))
32a1d 22 . . . 4 (φ → (x A → (φ ψ)))
41, 3alrimi 1396 . . 3 (φx(x A → (φ ψ)))
5 df-ral 2289 . . . 4 (x A ψx(x Aψ))
6 olc 619 . . . . . 6 (ψ → (φ ψ))
76imim2i 12 . . . . 5 ((x Aψ) → (x A → (φ ψ)))
87alimi 1324 . . . 4 (x(x Aψ) → x(x A → (φ ψ)))
95, 8sylbi 114 . . 3 (x A ψx(x A → (φ ψ)))
104, 9jaoi 623 . 2 ((φ x A ψ) → x(x A → (φ ψ)))
11 df-ral 2289 . 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 616  wal 1226  wnf 1329   wcel 1374  wral 2284
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 617  ax-5 1316  ax-gen 1318  ax-4 1381
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-ral 2289
This theorem is referenced by:  r19.32vr  2436
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