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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rspcegf | Structured version Visualization version GIF version | ||
| Description: A version of rspcev 3622 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
| Ref | Expression |
|---|---|
| rspcegf.1 | ⊢ Ⅎ𝑥𝜓 |
| rspcegf.2 | ⊢ Ⅎ𝑥𝐴 |
| rspcegf.3 | ⊢ Ⅎ𝑥𝐵 |
| rspcegf.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| rspcegf | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rspcegf.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 2 | rspcegf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
| 3 | 1, 2 | nfel 2920 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
| 4 | rspcegf.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
| 5 | 3, 4 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓) |
| 6 | eleq1 2829 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
| 7 | rspcegf.4 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 8 | 6, 7 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
| 9 | 1, 5, 8 | spcegf 3592 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) |
| 10 | 9 | anabsi5 669 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) |
| 11 | df-rex 3071 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
| 12 | 10, 11 | sylibr 234 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 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: rspcef 45077 stoweidlem46 46061 |
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