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Theorem rr-spce 41796
Description: Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.)
Hypotheses
Ref Expression
rr-spce.1 ((𝜑𝑥 = 𝐴) → 𝜓)
rr-spce.2 (𝜑𝐴𝑉)
Assertion
Ref Expression
rr-spce (𝜑 → ∃𝑥𝜓)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem rr-spce
StepHypRef Expression
1 rr-spce.2 . . . 4 (𝜑𝐴𝑉)
21elexd 3449 . . 3 (𝜑𝐴 ∈ V)
3 isset 3442 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
42, 3sylib 217 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
5 rr-spce.1 . . . 4 ((𝜑𝑥 = 𝐴) → 𝜓)
65ex 413 . . 3 (𝜑 → (𝑥 = 𝐴𝜓))
76eximdv 1920 . 2 (𝜑 → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜓))
84, 7mpd 15 1 (𝜑 → ∃𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wex 1782  wcel 2106  Vcvv 3429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3431
This theorem is referenced by:  grumnudlem  41884
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