Theorem rspesbcd 45920

Description: Restricted quantifier version of spesbcd 3829. (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 3803	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)
3	1, 2	sylibr 237	. . . 4 ⊢ (𝜑 → [𝐴 / 𝑥]𝑥 ∈ 𝐵)
4		rspesbcd.2	. . . 4 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)
5		sbcan 3787	. . . 4 ⊢ ([𝐴 / 𝑥](𝑥 ∈ 𝐵 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜓))
6	3, 4, 5	sylanbrc 595	. . 3 ⊢ (𝜑 → [𝐴 / 𝑥](𝑥 ∈ 𝐵 ∧ 𝜓))
7	6	spesbcd 3829	. 2 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
8		df-rex 3087	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
9	7, 8	sylibr 237	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086  [wsbc 3738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-v 3452  df-sbc 3739

This theorem is used by:  modelaxreplem3  45970