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Theorem rspesbcd 44935
Description: Restricted quantifier version of spesbcd 3891. (Contributed by Eric Schmidt, 29-Sep-2025.)
Hypotheses
Ref Expression
rspesbcd.1 (𝜑𝐴𝐵)
rspesbcd.2 (𝜑[𝐴 / 𝑥]𝜓)
Assertion
Ref Expression
rspesbcd (𝜑 → ∃𝑥𝐵 𝜓)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rspesbcd
StepHypRef Expression
1 rspesbcd.1 . . . . 5 (𝜑𝐴𝐵)
2 sbcel1v 3861 . . . . 5 ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
31, 2sylibr 234 . . . 4 (𝜑[𝐴 / 𝑥]𝑥𝐵)
4 rspesbcd.2 . . . 4 (𝜑[𝐴 / 𝑥]𝜓)
5 sbcan 3843 . . . 4 ([𝐴 / 𝑥](𝑥𝐵𝜓) ↔ ([𝐴 / 𝑥]𝑥𝐵[𝐴 / 𝑥]𝜓))
63, 4, 5sylanbrc 583 . . 3 (𝜑[𝐴 / 𝑥](𝑥𝐵𝜓))
76spesbcd 3891 . 2 (𝜑 → ∃𝑥(𝑥𝐵𝜓))
8 df-rex 3068 . 2 (∃𝑥𝐵 𝜓 ↔ ∃𝑥(𝑥𝐵𝜓))
97, 8sylibr 234 1 (𝜑 → ∃𝑥𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1775  wcel 2105  wrex 3067  [wsbc 3790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-v 3479  df-sbc 3791
This theorem is referenced by:  modelaxreplem3  44944
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