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Theorem rexlimd2 3007
 Description: Version of rexlimd 3008 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
rexlimd2.1 𝑥𝜑
rexlimd2.2 (𝜑 → Ⅎ𝑥𝜒)
rexlimd2.3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
Assertion
Ref Expression
rexlimd2 (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Proof of Theorem rexlimd2
StepHypRef Expression
1 rexlimd2.1 . . 3 𝑥𝜑
2 rexlimd2.3 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
31, 2ralrimi 2940 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
4 rexlimd2.2 . . 3 (𝜑 → Ⅎ𝑥𝜒)
5 r19.23t 3003 . . 3 (Ⅎ𝑥𝜒 → (∀𝑥𝐴 (𝜓𝜒) ↔ (∃𝑥𝐴 𝜓𝜒)))
64, 5syl 17 . 2 (𝜑 → (∀𝑥𝐴 (𝜓𝜒) ↔ (∃𝑥𝐴 𝜓𝜒)))
73, 6mpbid 221 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-ral 2901  df-rex 2902 This theorem is referenced by:  rexlimd  3008  sbcrext  3478  sbcrextOLD  3479
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