Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspcegf | Structured version Visualization version GIF version |
Description: A version of rspcev 3527 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 2911 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
4 | rspcegf.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfan 1907 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓) |
6 | eleq1 2818 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
7 | rspcegf.4 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
8 | 6, 7 | anbi12d 634 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
9 | 1, 5, 8 | spcegf 3497 | . . 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 3057 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
12 | 10, 11 | sylibr 237 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 Ⅎwnf 1791 ∈ wcel 2112 Ⅎwnfc 2877 ∃wrex 3052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-rex 3057 df-v 3400 |
This theorem is referenced by: rspcef 42234 stoweidlem46 43205 |
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