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Theorem rspcevOLD 3544
 Description: Obsolete version of rspce 3532 as of 12-Dec-2023. Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
rspcv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspcevOLD ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rspcevOLD
StepHypRef Expression
1 nfv 1915 . 2 𝑥𝜓
2 rspcv.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2rspce 3532 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3071 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rex 3076  df-v 3411 This theorem is referenced by: (None)
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