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Theorem rr-spce 42569
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 3467 . . 3 (𝜑𝐴 ∈ V)
3 isset 3460 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
42, 3sylib 217 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
5 rr-spce.1 . . . 4 ((𝜑𝑥 = 𝐴) → 𝜓)
65ex 414 . . 3 (𝜑 → (𝑥 = 𝐴𝜓))
76eximdv 1921 . 2 (𝜑 → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜓))
84, 7mpd 15 1 (𝜑 → ∃𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3447
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449
This theorem is referenced by:  grumnudlem  42657
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