Theorem rspcef 45820

Description: Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
rspcef.1	⊢ Ⅎ𝑥𝜓
rspcef.2	⊢ Ⅎ𝑥𝐴
rspcef.3	⊢ Ⅎ𝑥𝐵
rspcef.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rspcef	⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Proof of Theorem rspcef

Step	Hyp	Ref	Expression
1		rspcef.1	. 2 ⊢ Ⅎ𝑥𝜓
2		rspcef.2	. 2 ⊢ Ⅎ𝑥𝐴
3		rspcef.3	. 2 ⊢ Ⅎ𝑥𝐵
4		rspcef.4	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	rspcegf 45771	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569  Ⅎwnf 1812   ∈ wcel 2142  Ⅎwnfc 2909  ∃wrex 3088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rex 3089

This theorem is used by:  iinssdf  45885  rspced  45913  opnvonmbllem1  47374  smfresal  47530  smfmullem2  47534