Theorem rspced 45777

Description: Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 15-Feb-2025.)

Hypotheses

Ref	Expression
rspced.1	⊢ Ⅎ𝑥𝜒
rspced.2	⊢ Ⅎ𝑥𝐴
rspced.3	⊢ Ⅎ𝑥𝐵
rspced.4	⊢ (𝜑 → 𝐴 ∈ 𝐵)
rspced.5	⊢ (𝜑 → 𝜒)
rspced.6	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rspced	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Proof of Theorem rspced

Step	Hyp	Ref	Expression
1		rspced.4	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		rspced.5	. 2 ⊢ (𝜑 → 𝜒)
3		rspced.1	. . 3 ⊢ Ⅎ𝑥𝜒
4		rspced.2	. . 3 ⊢ Ⅎ𝑥𝐴
5		rspced.3	. . 3 ⊢ Ⅎ𝑥𝐵
6		rspced.6	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
7	3, 4, 5, 6	rspcef 45684	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜒) → ∃𝑥 ∈ 𝐵 𝜓)
8	1, 2, 7	syl2anc 595	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567  Ⅎwnf 1810   ∈ wcel 2149  Ⅎwnfc 2916  ∃wrex 3095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rex 3096

This theorem is referenced by:  rexanuz2nf  46098