Theorem rspcef 46010

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 45961	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145  Ⅎwnfc 2907  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rex 3087

This theorem is used by:  iinssdf  46075  rspced  46103  opnvonmbllem1  47564  smfresal  47720  smfmullem2  47724