Theorem rspced 45945

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 45852	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜒) → ∃𝑥 ∈ 𝐵 𝜓)
8	1, 2, 7	syl2anc 596	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  Ⅎwnfc 2912  ∃wrex 3091

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

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 2757  df-clel 2840  df-nfc 2914  df-rex 3092

This theorem is used by:  rexanuz2nf  46266