Theorem rspesbcd 45394

Description: Restricted quantifier version of spesbcd 3816. (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 3789	. . . . 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 3773	. . . 4 ⊢ ([𝐴 / 𝑥](𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜓))
6	3, 4, 5	sylanbrc 590	. . 3 ⊢ (𝜑 → [𝐴 / 𝑥](𝑥 ∈ 𝐵 ∧ 𝜓))
7	6	spesbcd 3816	. 2 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
8		df-rex 3066	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
9	7, 8	sylibr 236	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ∧ wa 397  ∃wex 1787   ∈ wcel 2121  ∃wrex 3065  [wsbc 3724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-v 3435  df-sbc 3725

This theorem is referenced by:  modelaxreplem3  45437