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Theorem r19.36av 2455
 Description: One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. In classical logic, the converse would hold if A has at least one element, but in intuitionistic logic, that is not a sufficient condition. (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 r19.35-1 2454 . 2 (x A (φψ) → (x A φx A ψ))
2 idd 21 . . . 4 (x A → (ψψ))
32rexlimiv 2421 . . 3 (x A ψψ)
43imim2i 12 . 2 ((x A φx A ψ) → (x A φψ))
51, 4syl 14 1 (x A (φψ) → (x A φψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301 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-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305  df-rex 2306 This theorem is referenced by:  iinss  3699
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