Theorem rspcef 42620

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 42566	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rex 3070  df-v 3434

This theorem is referenced by:  iinssdf  42688  opnvonmbllem1  44170  smfresal  44322  smfmullem2  44326