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Theorem rspesbcd 44951
Description: Restricted quantifier version of spesbcd 3882. (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 3855 . . . . 5 ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
31, 2sylibr 234 . . . 4 (𝜑[𝐴 / 𝑥]𝑥𝐵)
4 rspesbcd.2 . . . 4 (𝜑[𝐴 / 𝑥]𝜓)
5 sbcan 3837 . . . 4 ([𝐴 / 𝑥](𝑥𝐵𝜓) ↔ ([𝐴 / 𝑥]𝑥𝐵[𝐴 / 𝑥]𝜓))
63, 4, 5sylanbrc 583 . . 3 (𝜑[𝐴 / 𝑥](𝑥𝐵𝜓))
76spesbcd 3882 . 2 (𝜑 → ∃𝑥(𝑥𝐵𝜓))
8 df-rex 3070 . 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 1779  wcel 2108  wrex 3069  [wsbc 3787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-v 3481  df-sbc 3788
This theorem is referenced by:  modelaxreplem3  44988
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