Theorem rspcef 45179

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 45130	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2111  Ⅎwnfc 2879  ∃wrex 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rex 3057

This theorem is referenced by:  iinssdf  45246  rspced  45274  opnvonmbllem1  46740  smfresal  46896  smfmullem2  46900