Theorem rspesbcd 45518

Description: Restricted quantifier version of spesbcd 3838. (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 3811	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)
3	1, 2	sylibr 236	. . . 4 ⊢ (𝜑 → [𝐴 / 𝑥]𝑥 ∈ 𝐵)
4		rspesbcd.2	. . . 4 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)
5		sbcan 3795	. . . 4 ⊢ ([𝐴 / 𝑥](𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜓))
6	3, 4, 5	sylanbrc 592	. . 3 ⊢ (𝜑 → [𝐴 / 𝑥](𝑥 ∈ 𝐵 ∧ 𝜓))
7	6	spesbcd 3838	. 2 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
8		df-rex 3089	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
9	7, 8	sylibr 236	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1801   ∈ wcel 2144  ∃wrex 3088  [wsbc 3746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-v 3458  df-sbc 3747

This theorem is referenced by:  modelaxreplem3  45561