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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspced | Structured version Visualization version GIF version |
Description: Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 15-Feb-2025.) |
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)) |
Ref | Expression |
---|---|
rspced | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
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 44909 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜒) → ∃𝑥 ∈ 𝐵 𝜓) |
8 | 1, 2, 7 | syl2anc 583 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 Ⅎwnf 1781 ∈ wcel 2103 Ⅎwnfc 2888 ∃wrex 3072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1540 df-ex 1778 df-nf 1782 df-cleq 2726 df-clel 2813 df-nfc 2890 df-rex 3073 |
This theorem is referenced by: rexanuz2nf 45343 |
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