Theorem rspced 45204

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 45109	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜒) → ∃𝑥 ∈ 𝐵 𝜓)
8	1, 2, 7	syl2anc 584	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = 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:  rexanuz2nf  45530