Theorem rspcef 44964

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 44913	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  Ⅎwnfc 2893  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rex 3077

This theorem is referenced by:  iinssdf  45031  rspced  45063  opnvonmbllem1  46543  smfresal  46699  smfmullem2  46703