Theorem rspesbcd 44934

Description: Restricted quantifier version of spesbcd 3849. (Contributed by Eric Schmidt, 29-Sep-2025.)

Hypotheses

Ref	Expression
rspesbcd.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
rspesbcd.2	⊢ (𝜑 → [𝐴 / 𝑥]𝜓)

Assertion

Ref	Expression
rspesbcd	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rspesbcd

Step	Hyp	Ref	Expression
1		rspesbcd.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		sbcel1v 3822	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)
3	1, 2	sylibr 234	. . . 4 ⊢ (𝜑 → [𝐴 / 𝑥]𝑥 ∈ 𝐵)
4		rspesbcd.2	. . . 4 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)
5		sbcan 3806	. . . 4 ⊢ ([𝐴 / 𝑥](𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜓))
6	3, 4, 5	sylanbrc 583	. . 3 ⊢ (𝜑 → [𝐴 / 𝑥](𝑥 ∈ 𝐵 ∧ 𝜓))
7	6	spesbcd 3849	. 2 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
8		df-rex 3055	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
9	7, 8	sylibr 234	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  ∃wrex 3054  [wsbc 3756

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-v 3452  df-sbc 3757

This theorem is referenced by:  modelaxreplem3  44977