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Theorem rspceb2dv 3610
Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by Zhi Wang, 28-Sep-2024.)
Hypotheses
Ref Expression
rspceb2dv.1 ((𝜑𝑥𝐵) → (𝜓𝜒))
rspceb2dv.2 ((𝜑𝜒) → 𝐴𝐵)
rspceb2dv.3 ((𝜑𝜒) → 𝜃)
rspceb2dv.4 (𝑥 = 𝐴 → (𝜓𝜃))
Assertion
Ref Expression
rspceb2dv (𝜑 → (∃𝑥𝐵 𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥   𝜑,𝑥   𝜃,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspceb2dv
StepHypRef Expression
1 rspceb2dv.1 . . 3 ((𝜑𝑥𝐵) → (𝜓𝜒))
21rexlimdva 3149 . 2 (𝜑 → (∃𝑥𝐵 𝜓𝜒))
3 rspceb2dv.2 . . . 4 ((𝜑𝜒) → 𝐴𝐵)
4 rspceb2dv.3 . . . 4 ((𝜑𝜒) → 𝜃)
5 rspceb2dv.4 . . . . 5 (𝑥 = 𝐴 → (𝜓𝜃))
65rspcev 3606 . . . 4 ((𝐴𝐵𝜃) → ∃𝑥𝐵 𝜓)
73, 4, 6syl2anc 583 . . 3 ((𝜑𝜒) → ∃𝑥𝐵 𝜓)
87ex 412 . 2 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
92, 8impbid 211 1 (𝜑 → (∃𝑥𝐵 𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wrex 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065
This theorem is referenced by:  ipolubdm  47886  ipoglbdm  47889
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