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Mirrors > Home > MPE Home > Th. List > r19.12 | Structured version Visualization version GIF version |
Description: Restricted quantifier version of 19.12 2337. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2381, ax-ext 2790. (Revised by Wolf Lammen, 17-Jun-2023.) |
Ref | Expression |
---|---|
r19.12 | ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3141 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑)) | |
2 | nfv 1906 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
3 | nfra1 3216 | . . . . 5 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑 | |
4 | 2, 3 | nfan 1891 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑) |
5 | 4 | nfex 2334 | . . 3 ⊢ Ⅎ𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑) |
6 | 1, 5 | nfxfr 1844 | . 2 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
7 | ax-1 6 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)) | |
8 | rsp 3202 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑)) | |
9 | 8 | com12 32 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 𝜑 → 𝜑)) |
10 | 9 | reximdv 3270 | . . 3 ⊢ (𝑦 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
11 | 7, 10 | sylcom 30 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝜑)) |
12 | 6, 11 | ralrimi 3213 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1771 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-ral 3140 df-rex 3141 |
This theorem is referenced by: iuniin 4922 ucncn 22821 ftc1a 24561 heicant 34808 rngoid 35061 rngmgmbs4 35090 intimass 39877 intimag 39879 |
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