Theorem rspced 43862

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 43759	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜒) → ∃𝑥 ∈ 𝐵 𝜓)
8	1, 2, 7	syl2anc 585	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542  Ⅎwnf 1786   ∈ wcel 2107  Ⅎwnfc 2884  ∃wrex 3071

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rex 3072  df-v 3477

This theorem is referenced by:  rexanuz2nf  44203