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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rspcef | Structured version Visualization version GIF version | ||
| Description: Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| Ref | Expression |
|---|---|
| rspcef.1 | ⊢ Ⅎ𝑥𝜓 |
| rspcef.2 | ⊢ Ⅎ𝑥𝐴 |
| rspcef.3 | ⊢ Ⅎ𝑥𝐵 |
| rspcef.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| rspcef | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rspcef.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 2 | rspcef.2 | . 2 ⊢ Ⅎ𝑥𝐴 | |
| 3 | rspcef.3 | . 2 ⊢ Ⅎ𝑥𝐵 | |
| 4 | rspcef.4 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 5 | 1, 2, 3, 4 | rspcegf 45028 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ∃wrex 3070 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rex 3071 |
| This theorem is referenced by: iinssdf 45144 rspced 45172 opnvonmbllem1 46647 smfresal 46803 smfmullem2 46807 |
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